5.1: Angles and Radian Measure Date: Pre-Calculus

5.1: Angles and Radian Measure Date: ________

Pre-Calculus

*Use Section 5.1 (beginning on pg. 482) to complete the following

Trigonometry: “measurement of triangles”

An angle is formed by two rays that have a common endpoint. One ray is called __________ side and the other the

_______________ side.

Draw and label the diagram:

An angle is in standard position if:

Its vertex is at the origin of a rectangular coordinate system

Its initial side lies along the positive x-axis

Draw and label both diagrams:

Positive angles are generated by a counterclockwise rotation. Negative angles are generated by a clockwise rotation. If the terminal side of an angle lies in a specific quadrant (I, II, III, IV), we say that the angle lies in that quadrant. If the terminal side of an angle lies on the x-axis or the y-axis, the angle is called a quadrantal angle. Measuring Angles Using Degrees

Think about a clock. One full rotation is 60 minutes and this is a rotation of 360°.

Fractional parts of degrees are measured in minutes and seconds

One minute, written 1’ is _____ degree: 1’ = _______.

One second, written 1’’ is _________ degree: 1’’ = ____________.

For example, 31°47′12′′ = (31 +47

60+

12

3600) ° ≈ 31.787°.

Many calculators have keys for changing an angle from degree-minute-second notation (𝐷°𝑀′𝑆′′) to a decimal form

and vice versa.

Measuring Angles Using Radians:

Use the circle at the right:

1. Create an angle with the vertex at the center of the circle (a central angle)

2. Label the sides of the angle, r, for the radius

If the central angle intercepts an arc along the circle measuring r units, the measure of such an angle is 1 radian

3. Label the arc created by the central angle r

4. Label the central angle 1 radian

*1 radian = when arc and radius are equal

The radian measure of any central angle is the length of the intercepted arc divided by the circle’s radius.

Construct an angle 𝛽 with measure 2 radians.

Radian Measure: Consider an arc of length s on a circle of radius r. The measure of the central angle, 𝜃 (“theta”), that intercepts the arc is

𝜃 =𝑠

𝑟 radians.

Relationship between Degrees and Radians: Compare the number of degrees and the number of radians in one complete rotation.

𝜃 =𝑠

𝑟=

2𝜋𝑟

𝑟= 2𝜋

So, 360° = 2𝜋 radians. Thus, 180° = 𝜋 radians.

Conversion between Degrees and Radians:

Using the basic relationship that 180° = 𝜋 radians,

1. To convert degrees to radians, multiply degrees by _________________________

2. To convert radian to degrees, multiply radians by ___________________________

Angles that are fractions of a complete rotation are usually expressed in radian measure as factional multiples of 𝜋

rather than as decimal approximations. For example 𝜃 =𝜋

2.

s

r

r

𝜃

Ex. 2: Convert each angle in degrees to radians:

a) 60° b) 270° c) −300°

Ex. 3: Convert each angle in radians to degrees:

a) 𝜋

4 radians b) −

4𝜋

3 radians c) 6 radians

Drawing Angles in Standard Position:

*Goal: Learn to THINK in RADIANS

Ex. 4: Draw and label each angle in standard position.

a) 𝜃 = −𝜋

4 b) 𝛼 =

3𝜋

4 c) 𝛽 = −

7𝜋

4 d) 𝛾 =

13𝜋

4

Terminal Side Radian Measure of Angle Degree Measure of Angle

1

12 revolution

1

8 revolution

1

6 revolution

1

4 revolution

1

3 revolution

1

2 revolution

2

3 revolution

3

4 revolution

7

8 revolution

1 revolution

Coterminal Angles- two angles with the same ________________ and _____________________ sides.

Every angle has infinitely many coterminal angles.

Increasing or decreasing the degree measure of an angle in standard position by an integer multiple of 360° results

in a coterminal angle. Thus, an angle of 𝜃° is coterminal with angles of 𝜃° ± 360°𝑘, where k is an integer.

Increasing or degreasing the radian measure of an angle by an integer multiple of 2𝜋 results in a coterminal angle.

Thus, an angle of 𝜃 radians is coterminal with angles of 𝜃 ± 2𝜋𝑘, where k is an integer.

Ex. 5: Find a positive angle less than 360° that is coterminal with each of the following:

a) a 400° angle b) a −135° angle

Ex. 6: Find a positive angle less than 2𝜋 that is coterminal with each of the following:

a) a 13𝜋

5 angle b) a −

𝜋

15 angle

Ex. 7: Find a positive angle less than 360° or 2𝜋 that is coterminal with each of the following:

a) an 855° angle b) a 17𝜋

3 angle c) a −

25𝜋

6 angle

5.1 Continued: Angles and Radian Measure Date: ________ Pre-Calculus The Length of a Circular Arc: Ex 1: The minute hand of a clock is 6 inches long and moves from 12 to 4 o’clock. How far does the tip of the minute hand move? Express your answer in terms of 𝜋 and then round to two decimal places. Ex 2: The figure shows a highway sign that warns of a railway crossing. The lines that form the cross pass through the circle’s center and intersect at right angles. If the radius of the circle is 24 inches, find the length of each of the four arcs formed by the cross. Express your answer in terms of 𝜋 and then round to two decimal places. Ex 3: A circle has a radius of 6 inches. Find the length of the arc intercepted by a central angle of 45°. Express arc

length in terms of 𝜋. Then round your answer to two decimal places.

Linear and Angular Speed: If a point is in motion on a circle of radius r through an angle of 𝜃 radians in time t, then its LINEAR SPEED is: and its ANGULAR SPEED is: Note: The linear speed is the product of the radius and the angular speed, where w is the angular speed in radians per unit of time.

Ex 4: The angular speed of a point on Earth is 𝜋

12 radian per hour. The Equator lies on a circle of radius

approximately 4000 miles. Find the linear velocity, in miles per hour, of a point on the Equator.

Ex 5: A Ferris wheel has a radius of 25 feet. The wheel is rotating at two revolutions per minute. Find the linear

speed, in feet per minute, of a seat on this Ferris wheel.

Ex 6: A water wheel has a radius of 12 feet. The wheel is rotating at 20 revolutions per minute. Find the linear

speed, in feet per minute, of the water.

Ex. 9: Long before iPods that hold thousands of songs and play them with superb audio quality, individual songs

were delivered on 75-rpm and 45-rpm circular records. A 45-rpm record has an angular speed of 45 revolutions per

minute. Find the linear speed, in inches per minute, at the point where the needle is 1.5 inches from the record’s

center.

Homework: MathXL 5.1 Homework

5.2: Right Triangle Trigonometry Date: ________ Pre-Calculus Fundamental Identities: Trigonometry Identities: Quotient Identities: Pythagorean Theorem: Divide both sides by 𝑐2 What do you observe? Pythagorean Theorem Identities:

Ex 1: Given that 𝑠𝑖𝑛𝜃 =1

2 and 𝜃 is an acute angle, find the value of 𝑐𝑜𝑠𝜃 using a trig identity.

Consider the relationship between the 𝑠𝑖𝑛30° = _______________, 𝑐𝑜𝑠60°= _________________.

So, 𝑠𝑖𝑛𝜃 = cos( )

Any pair of trigonometric functions f and g for which 𝑓(𝜃) = 𝑔(90° − 𝜃) are called __________________.

Cofunction Identities: If 𝜃 is measured in radians, replace the 90° with 𝜋

2.

The value of a trigonometric function of 𝜃 is equal to the cofunction of the complement of 𝜃. Cofunctions of

complementary angles are equal.

Ex 2: Find a cofunction with the same value as the given expression:

a) 𝑠𝑖𝑛46° b) 𝑐𝑜𝑡𝜋

12

Using a calculator to evaluate trigonometric functions:

Ex 3: Using a calculator to find the value to four decimal places:

a) 𝑠𝑖𝑛72.8° b) 𝑐𝑠𝑐1.5

Applications:

The angle formed by a horizontal line and the line of sight to an object that is above the horizontal line is called the

_____________ of _________________________.

The angle formed by a horizontal line and the line of sight to an object that is below the horizontal line is called the

_____________ of _________________________.

Ex. 4: A flagpole that is 14 meters tall casts a shadow 10 meters long. Find the angle of elevation to the sun to the

nearest degree.

Ex. 5: Find the exact value of each expression. Do not use a calculator.

a) 𝑡𝑎𝑛

𝜋

4

2−

1

𝑠𝑒𝑐𝜋

6

b) 1 + 𝑠𝑖𝑛240° + 𝑠𝑖𝑛250° c) 𝑐𝑜𝑠12°𝑠𝑖𝑛78° + 𝑐𝑜𝑠78°𝑠𝑖𝑛12°

Ex. 6: Express the exact value of the function as a single fraction. Do not use a calculator.

If 𝑓(𝜃) = 2𝑠𝑖𝑛𝜃 − 𝑠𝑖𝑛𝜃

2, find 𝑓 (

𝜋

3).

Ex. 7: At a certain time of day, the angle of elevation of the sun is 40°. To the nearest foot, find the height of a tree

whose shadow is 35 feet long.

Ex. 8: The Washington Monument is 555 feet high. If you are standing one quarter of a mile from the base of the

monument and looking to the top, find the angle of elevation to the nearest degree.

Homework: Math XL: 5.2 Homework

5.3: Trigonometric Functions of Any Angle Date: ________

Pre-Calculus

Trigonometric Functions of Any Angle: The point P = (x,y) is a point r units away from the origin on the terminal side

of 𝜃. A right triangle is formed by drawing a line segment from P = (x,y) _____________________ to the x-axis.

Note: y is the length of the side opposite 𝜃 and x is the length of the side adjacent to 𝜃.

Definitions of Trigonometric Functions of Any Angle: Let 𝜃 be any angle in standard position and let P = (x,y) be a

point on the terminal side of 𝜃. If r = _____________________ is the _______________ from (0,0) to (x,y), the six

trigonometric functions of 𝜃 are defined by the following ratios:

Since P = (x,y) is any point on the terminal side of 𝜃 other than the origin, 𝑟 = ____________________ cannot be zero.

(Think of “r” as the radius….distance from the origin to P)

Ex 1: Let P = (1,-3) be a point on the terminal side of 𝜃. Find each of the six trigonometric functions of 𝜃. (Start by

drawing the angle in standard position.)

Trigonometric Functions of Quadrantal Angles:

Ex. 2: Find the values of the six trigonometric functions, if they exist, for each of the four quadrantal angles.

a) 𝜃 = 0° = 0 b) 𝜃 = 90° =𝜋

2 c) 𝜃 = 180° = 𝜋 d) 𝜃 = 270° =

3𝜋

2

The Signs of the Trigonometric Functions: If 𝜃 is not a quadrantal angle, the sign of a trigonometric function

depends on the quadrant in which 𝜃 lies.

In all four quadrants, r, is positive (radius).

Since 𝑠𝑖𝑛𝜃 =𝑦

𝑟, the sign of this ratio depends on y. Similarly the sign of the cosine ratio depends on x. Tangent is

the ratio of sine and cosine so it is dependent on the signs of ______ x and y.

*Think alphabetical … (x,y) … (cosine,sine)

Now, look at all 4 quadrants:

Your book’s phrase: A Smart Trig Class (see pg. 517)

Ex 3: If 𝑠𝑖𝑛𝜃 < 0 and 𝑐𝑜𝑠𝜃 < 0, name the quadrant in which angle 𝜃 lies.

Ex. 4: Given 𝑡𝑎𝑛𝜃 = −1 and 𝑐𝑜𝑠𝜃 < 0, find 𝑠𝑖𝑛𝜃 and 𝑠𝑒𝑐𝜃.

Reference Angles: Let 𝜃 be a nonacute angle in standard position that lies in a quadrant. Its _______________

___________ is the ______________ ____________ angle 𝜃′ (“__________ __________”) formed by the

terminal side of 𝜃 and the x-axis.

a) 𝜃 is in Quadrant II b) 𝜃 is in Quadrant III c) 𝜃 is in Quadrant IV

Ex. 5: Find the reference angle, 𝜃′, for each of the following angles:

a) 𝜃 = 210° b) 𝜃 =7𝜋

4 c) 𝜃 = −240° d) 𝜃 = 3.6

Finding Reference Angles for Angles Greater than 360° (2𝜋) or less than −360° (−2𝜋):

1) Find a positive angle 𝛼 less than 360° or 2𝜋 that is ________________ with the given angle.

2) Draw 𝛼 is standard position.

3) Use the drawing to find the reference angle for the given angle. The positive acute angle formed by the

terminal side of 𝛼 and the x-axis is the _________________ _____________.

Ex. 6: Find the reference angle for each of the following angles:

a) 𝜃 = 665° b) 𝜃 =15𝜋

4 c) 𝜃 = −

11𝜋

3

Evaluating Trigonometric Functions Using Reference Angles:

The value of a trigonometric function of any angle 𝜃 is found as follows:

1. Find the associated reference angle, 𝜃′, and the function value for 𝜃′.

2. Use the quadrant in which 𝜃 lies to prefix the appropriate sign to the function value in step 1.

Ex. 7: Use reference angles to find the exact value of the following trigonometric functions:

1. 𝑠𝑖𝑛300° b) 𝑡𝑎𝑛5𝜋

4 c) 𝑠𝑒𝑐 (−

𝜋

6)

Ex. 8: Use reference angles to find the exact value of each of the following trigonometric functions:

a) 𝑐𝑜𝑠17𝜋

6 b) sin (−

22𝜋

3)

Homework: pg. 525 #3-84 (multiples of 3)---Use page 524 as a reference.

5.4: Trigonometric Functions of Real Numbers; Periodic Functions Date: ________

Pre-Calculus

Unit Circle:

The equation of this unit circle is: ___________________________.

Recall: 𝑠 = 𝑟𝜃, so in the unit circle, where the central angle measures t radians, 𝑠 = ________.

So, in a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc.

𝑥 = ___________ , 𝑦 = ___________

Definitions of the Trigonometric Functions in Terms of a Unit Circle

If t is a real number and P=(x,y) is a point on the unit circle that corresponds to t, then

𝑠𝑖𝑛𝑡 = _______ 𝑐𝑠𝑐𝑡 = _________ , 𝑦 ≠ 0

𝑐𝑜𝑠𝑡 = _______ 𝑠𝑒𝑐𝑡 = _________ , 𝑥 ≠ 0

𝑡𝑎𝑛𝑡 = ________ , 𝑥 ≠ 0 𝑐𝑜𝑡𝑡 = _________ , 𝑦 ≠ 0

Note: 𝑥2 + 𝑦2 = 1 is equivalent to the Pythagorean identity 𝑐𝑜𝑠2𝑥 + 𝑠𝑖𝑛2𝑥 = 1.

Ex. 1: Find the values of the trig functions at t if 𝑃 = (√3

2,1

2).

Ex. 2: Use a unit circle to find the values of the trigonometric functions at 𝑡 = 𝜋.

Domain and Range of Sine and Cosine Functions

The domain of the sine function and the cosine function is (−∞,∞), the set of all real numbers. The range of these

functions is [−1,1], the set of all real numbers from -1 to 1, inclusive.

Even and Odd Trigonometric Functions

Even: 𝑓(−𝑡) = ___________ Odd: 𝑓(−𝑡) = _____________

If P(cost,sint); Q((cos(-t),sin(-t)) The x-coordinates of P and Q are the same. Thus, cos(−𝑡) = ____________ , so cosine is an _________ function.

The y-coordinates of P and Q are opposites of each other. Thus sin(−𝑡) = ______________ , so sine is an ________

function.

EVEN FUNCTIONS: ODD FUNCTIONS:

Ex. 3: Find the exact value of each trigonometric function:

a) cos (−60°) b) tan (−𝜋

6)

A function f is periodic if there exists a positive number p such that 𝑓(𝑡 + 𝑝) = 𝑓(𝑡) for all t in the domain of f. The

smallest positive number p for which f is periodic is called the period of f.

Periodic Properties of the Sine and Cosine Functions:

sin(𝑡 + 2𝜋) = 𝑠𝑖𝑛𝑡 and cos(𝑡 + 2𝜋) = 𝑐𝑜𝑠𝑡

The sine and cosine functions are periodic functions and have a period of ______. (Similarly, cosecant and secant

have a period of 2𝜋.)

Ex. 4: Find the exact value of each trigonometric function:

a) 𝑐𝑜𝑠405° b) 𝑠𝑖𝑛7𝜋

3

Periodic Properties of the Tangent and Cotangent Functions

tan(𝑡 + 𝜋) = 𝑡𝑎𝑛𝑡 and cot(𝑡 + 𝜋) = 𝑐𝑜𝑡𝜋

The tangent and cotangent functions are periodic functions and have period __________

Repetitive Behavior of the Sine, Cosine, and Tangent Functions:

For any integer n and real number t,

sin(𝑡 + 2𝜋𝑛) = _____________

cos(𝑡 + 2𝜋𝑛) = __________________

tan(𝑡 + 𝜋𝑛) = ____________

Homework: pg. 532 #1 – 18, 20 – 36(e), 44 – 46

5.5: Graphs of Sine and Cosine Date: ________ Pre-Calculus Intro to Sine and Cosine Functions:

𝑦 = 𝑠𝑖𝑛𝑥

Amplitude: __________

Period: ___________

Scaling: __________

𝑦 = 𝑐𝑜𝑠𝑥

Amplitude: __________

Period: ____________

Scaling: _________

Graphing sine and cosine functions in general:

𝑦 = 𝐴𝑠𝑖𝑛(𝐵𝑥 − 𝐶) + 𝐷 and 𝑦 = 𝐴𝑐𝑜𝑠(𝐵𝑥 − 𝐶) + 𝐷

1. |A| = amplitude (If A<0, the graph is reflected over the x-axis.)

2. Period = 2𝜋

𝐵

3. Horizontal shift (called the _________ ________): The starting point changes from 𝑥 = 0 to 𝑥 =𝐶

𝐵

(Find this by setting 𝐵𝑥 − 𝐶 = 0.) If 𝐶

𝐵> 0, the shift is to the right. If

𝐶

𝐵< 0, the shift is to the left. The

number 𝐶

𝐵 is called the phase shift.

4. Vertical shift: The constant D causes vertical shifts in the graph. If D is positive, the shift is D units

__________. If D is negative, the shift is |D| units _____________. These vertical shifts result in graphs that

oscillate about the horizontal line ________ rather than the x-axis. Thus, the maximum y is D + |A| and

the minimum y is D - |A|. (These values give the _________ of the function.)

Examples: Graph each function. Draw two cycles. Label the max/min points and intercepts (5 key points per cycle).

Ex 1: 𝑦 = 3cos (𝑥 + 𝜋)

Amplitude: _______

Period: _______

Scale by: ________

Vertical shift: __________

Phase shift: ____________

Ex 2: 𝑦 = 2 − sin (2𝜋

3𝑥)

Amplitude: _______

Period: _______

Scale by: ________

Vertical shift: __________

Phase shift: ____________

Ex 3: 𝑦 =2

3𝑐𝑜𝑠 (

𝑥

2−

𝜋

4)

Amplitude: _______

Period: _______

Scale by: ________

Vertical shift: __________

Phase shift: ____________

Ex 4: 𝑦 = 3 cos(𝑥 + 𝜋) − 3

Amplitude: _______

Period: _______

Scale by: ________

Vertical shift: __________

Phase shift: ____________

Homework: pg. 554 #13, 20, 23, 26, 41, 44, 47, 50, 62, 64, 66, 84 – 87

5.6: Graphs of Other Trig Functions (Tangent and Cotangent) Date: ________ Pre-Calculus

𝑦 = tan 𝑥

Period: ______

VA: ___________

Domain: ____________

Range: ____________

x-intercepts:

Symmetry:

Graphing Variations of 𝑦 = tan 𝑥 in the form 𝑦 = 𝐴 tan(𝐵𝑥 − 𝐶)

1. Period = _______

2. Find two consecutive asymptotes by finding an interval containing one period: _______________________

The asymptotes occur at:

a.

b.

3. Identify an x-intercept, midway between the consecutive asymptotes

4. Amplitude = ______

Ex 1: 𝑦 = tan (𝜋 − 𝑥)

Amplitude: ___________

Period: ___________

Asymptotes:

𝑦 = 𝑐𝑜𝑡𝑥

Period: ______

VA: ___________

Domain: ____________

Range: ____________

x-intercepts:

Symmetry:

Graphing Variations of 𝑦 = cot 𝑥 in the form 𝑦 = 𝐴 cot(𝐵𝑥 − 𝐶)

1. Period = _______

2. Find two consecutive asymptotes by finding an interval containing one period: _______________________

The asymptotes occur at:

a.

b.

3. Identify an x-intercept, midway between the consecutive asymptotes

4. Amplitude = ______

Ex 2: 𝑦 = cot (𝑥

2)

Amplitude: ___________

Period: ___________

Asymptotes:

Ex 3: 𝑦 = −2cot (𝑥 +𝜋

2)

Amplitude: ___________

Period: ___________

Phase Shift: ____________

Asymptotes:

Ex 4: 𝑦 =1

2cot 2𝑥

Amplitude: ___________

Period: ___________

Asymptotes:

Ex 5: 𝑦 = −2 tan1

2𝑥

Amplitude: ___________

Period: ___________

Asymptotes:

Homework: pg. 567 #1 – 4, 6 – 12(e), 13 – 16, 18 – 24(e), 45, 46, 59

5.6 Continued: Graphs of Other Trigonometric Functions Date: ________ Pre-Calculus

𝑦 = sec 𝑥 𝑦 = 𝑐𝑠𝑐𝑥

Reciprocal function:

Domain:

Range:

Period:

Vertical Asymptotes:

Symmetry:

𝑦 = sec 𝑥 𝑦 = csc 𝑥

Ex 1: 𝑦 = 3 csc𝑥

2

Amplitude: ________

Period: _______

Scale by: _______

Ex 2: 𝑦 = −1

2sec 𝑥

Amplitude: ________

Period: _______

Scale by: _______

Ex 3: 𝑦 = csc𝑥

3

Amplitude: ________

Period: _______

Scale by: _______

Ex 4: 𝑦 = sec(𝜋𝑥) − 1

Amplitude: ________

Period: _______

Scale by: _______

Ex 5: 𝑦 = 2 csc (𝑥 +𝜋

4)

Amplitude: ________

Period: _______

Scale by: _______

Phase shift: ________________

Homework: pg. 568 #26 – 44(e), 47, 48, 61, 93 – 96

5.7: Inverse Trigonometric Functions Date: ________ Pre-Calculus Recall: In order for a function to have an inverse function, the graph of the function must pass the

_________________ ________ _________. On their entire domain, the trig functions do ______ pass this test.

INVERSE SINE

Restrict 𝑦 = 𝑠𝑖𝑛𝑥 so that _______________________.

Then it will have an inverse function.

The inverse function for 𝑠𝑖𝑛𝑥 is _____________________ or _________________________.

(For inverse functions, the domain/range are switched.)

Sine is an __________ function - symmetry with the _____________________.

Note: 𝑦 = 𝑠𝑖𝑛−1𝑥 ≠1

𝑠𝑖𝑛𝑥.

Graphically inverses are reflections over the line ________________. (true for exp/log functions)

𝜃 𝑠𝑖𝑛𝜃

Ex 1: Find the exact value of each of the following:

a) 𝑠𝑖𝑛−1 (√3

2) = ______________ b) 𝑠𝑖𝑛−1 (−

√2

2) = __________________

INVERSE COSINE

Restrict 𝑦 = 𝑐𝑜𝑠𝑥 so that _______________________.

Then it will have an inverse function.

The inverse function for 𝑐𝑜𝑠𝑥 is _____________________ or _________________________.

(For inverse functions, the domain/range are switched.)

Cosine is an __________ function - symmetry with the _____________________.

𝜃 𝑐𝑜𝑠𝜃

Ex 2: Find the exact value of each of the following:

a) 𝑐𝑜𝑠−1 (−1

2) = ____________ b) 𝑐𝑜𝑠−1 (

√2

2) = ________________

INVERSE TANGENT

Restrict 𝑦 = 𝑡𝑎𝑛𝑥 so that _______________________.

Then it will have an inverse function.

The inverse function for 𝑡𝑎𝑛𝑥 is ____________________ or _________________________.

(For inverse functions, the domain/range are switched.)

Cosine is an __________ function - symmetry with the _____________________.

𝜃 𝑡𝑎𝑛𝜃

Ex. 3: Find the exact value of each of the following:

a) 𝑡𝑎𝑛−1(−1) = ___________ b) 𝑡𝑎𝑛−1 (√3

3) = ____________

Ex. 4: Use a calculator to find the value of each expression to four decimal places.

a) 𝑐𝑜𝑠−1 (1

3) = _______________ b) 𝑡𝑎𝑛−1(35.85) =________________ c) 𝑠𝑖𝑛−12 =__________

COMPOSITION OF FUNCTIONS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS

Recall: 𝑓(𝑓−1(𝑥)) = 𝑥 for ____________ and 𝑓−1(𝑓(𝑥)) = 𝑥 for _______________.

INVERSE PROPERTIES

sin(𝑠𝑖𝑛−1𝑥) = 𝑥 for ______________________

𝑠𝑖𝑛−1(𝑠𝑖𝑛𝑥) = 𝑥 for _______________________

cos(𝑐𝑜𝑠−1𝑥) = 𝑥 for ________________________

𝑐𝑜𝑠−1(𝑐𝑜𝑠𝑥) = 𝑥 for ________________________

tan(𝑡𝑎𝑛−1𝑥) = 𝑥 for _________________________

𝑡𝑎𝑛−1(𝑡𝑎𝑛𝑥) = 𝑥 for _________________________

Ex. 5: Find the exact value (if possible) for each of the following expressions:

a) cos(𝑐𝑜𝑠−10.7) = ______________________

b) 𝑠𝑖𝑛−1(𝑠𝑖𝑛𝜋) = ______________________

c) 𝑐𝑜𝑠[𝑐𝑜𝑠−1(−1.2)] = ____________________

EVALUATING A COMPOSITE TRIG EXPRESSION (Draw a triangle!)

Ex. 6: Find the exact value of sin (𝑡𝑎𝑛−1 (3

4)).

Ex. 7: Find the exact value of 𝑐𝑜𝑠 [𝑠𝑖𝑛−1 (−1

2)].

SIMPLIFYING EXPRESSIONS INVOLVING TRIG FUNCTIONS

Ex. 8: If 𝑥 > 0, write 𝑠𝑒𝑐(𝑡𝑎𝑛−1𝑥) as an algebraic expression in 𝑥.

Homework due Tuesday 12/13: Graphing Trig Functions Review (handout) & pg. 584 #1 – 30 (no calc for #1 – 18)

Quiz #8 (5.5 – 5.6) – NON CALCULATOR! Wednesday 12/14

Homework due Wednesday 12/14 (Section 5.7): pg. 584 #32 – 72(e), 95, 96, 116 – 121, 124 – 126 -- Make sure you

know the sin, cos, tan values in the tables!

5.8: Applications of Trigonometric Functions Date: ________ Pre-Calculus Objectives:

Solve a right triangle.

Solve problems involving bearings.

Solving Right Triangles—finding the missing lengths of the sides and angles of a triangle.

Generic diagram of a right triangle

Ex. 1: Let 𝐴 = 62.7° and 𝑎 = 8.4. Solve the right triangle, rounding lengths to two decimal places.

Ex. 2: from a point on level ground 80 feet from the base of the Eiffel Tower, the angle of elevation is 85.4°.

Approximate the height of the Eiffel Tower to the nearest foot.

Ex. 3: A guy wire (support wire) is 13.8 yards long and is attached from the ground to a pole 6.7 yards above the

ground. Find the angle, to the nearest tenth of a degree that the wire makes with the ground.

Using Two Right Triangles to Solve a Problem

Ex. 4: You are standing on level ground 800 feet from Mt. Rushmore, looking at the sculpture of Abraham Lincoln’s

face. The angle of elevation to the bottom of the sculpture is 32° and the angle of elevation to the top is 35°. Find

the height of the sculpture of Lincoln’s face to the nearest tenth of a foot.

Trigonometry and Bearings

The bearing from a point O to a point P is the acute angle, measured in degrees between 𝑂𝑃⃗⃗⃗⃗ ⃗ and a north-south line.

Ex. 5: Find the bearing from O to P in each diagram.

Ex. 6: A boat leaves the entrance to a harbor and travels 150 miles on a bearing of 𝑁 53°𝐸. How many miles north

and how many miles east form the harbor has the boat traveled?

Ex. 7: You leave your house and run 2 miles due west followed by 1.5 miles due north. At that time, what is your

bearing from your house?

Ex. 8: You leave the entrance to a system of hiking trails and hike 2.3 miles on a bearing of 𝑆 31° 𝑊. Then the trail

turns 90° and you hike 3.5 miles on a bearing of 𝑁 59° 𝑊. At that time:

a) How far are you, to the nearest tenth of a mile, from the entrance to the trail system?

b) What is your bearing, to the nearest tenth of a degree, from the entrance to the trail system?

Homework: pg. 595 #6, 10, 14, 16, 42 – 58(e) (We will continue section 5.8 tomorrow.)

5.8 Continued: Simple Harmonic Motion Date: ________ Pre-Calculus Simple Harmonic Motion—Consider a ball attached to a spring hung from the ceiling. You pull the ball down and

then release it. If we neglect the effects of friction and air resistance, the ball will continue bobbing up and down on

the end of the spring. These up-and down oscillations are called simple harmonic motion. The position of the ball

before you pull the ball down is d=0. This rest position is called the equilibrium position.

An object that moves on a coordinate axis is in simple harmonic motion if its distance from the origin, d, at time t, is

given by either

𝑑 = acos(𝜔𝑡) 𝑜𝑟 𝑑 = asin (𝜔𝑡)

Use 𝑑 = acos(𝜔𝑡) if ____________________________________________________________

Use 𝑑 = asin (𝜔𝑡) if _____________________________________________________________

The motion has _______________ |𝑎|, the maximum displacement of the object from its rest position.

The __________ of the motion is 2𝜋

𝜔, where 𝜔 > 0.

The period gives the time it takes for the motion to go through one _______________ ___________.

Ex. 10: A ball on a spring is pulled 6 inches below its rest position and then released. The period for the motion is 4

seconds. Write the equation for the ball’s simple harmonic motion.

Frequency of an Object in simple Harmonic Motion

An object in simple harmonic motion given by

𝑑 = acos(𝜔𝑡) 𝑜𝑟 𝑑 = asin (𝜔𝑡)

has frequency f, given by

𝑓 =𝜔

2𝜋, 𝜔 > 0.

Equivalently, 𝑓 =1

𝑝𝑒𝑟𝑖𝑜𝑑.

Ex. 11: An object moves in simple harmonic motion described by 𝑑 = 12𝑐𝑜𝑠𝜋

4𝑡, where t is measured in seconds

and d in centimeters. Find:

a) the maximum displacement:

b) the frequency:

c) the time required for one cycle:

Ex. 12: An object moves in simple harmonic motion described by 𝑑 = −1

2𝑠𝑖𝑛 (

𝜋

4𝑡 −

𝜋

2), where t is measured in

seconds and d in centimeters. Find:

a) the maximum displacement:

b) the frequency:

c) the time required for one cycle:

d) the phase shift of the motion:

Homework: pg. 596 #20, 24, 26, 37, 39, 60, 62, 70, 72 – 77; pg. 601 #18 – 20, pg. 603 #106 – 120(e)