Chapter 5 Trigonometric Functions. Section 1: Angles and Degree Measure The student will be able to: Convert decimal degree measures to degrees, minutes,

Chapter 5Trigonometric Functions

Section 1: Angles and Degree Measure

The student will be able to:• Convert decimal degree measures to degrees, minutes, and seconds and vice versa.• Find the number of degrees in a given number of rotations.• Identify angles that are coterminal with a given angle.

An angle may be generated by rotating one of two rays that share a fixed endpoint known as the vertex. One of the rays is fixed (doesn’t move) along the x axis and is called the initial side. The other side, which rotates, is called the terminal side.

If an angle is rotated in a counterclockwise direction, it is positive.If an angle is rotated in a clockwise direction, it is negative.

An angle is in standard position if its initial side is along the x axis.

We usually measure angles in degrees with decimnmals, but its also possible to measure them in degrees, minutes, and seconds.

Quadrant II

Quadrant III Quadrant IV

Quadrant 1

Terminal side of the angle

Terminal side of the angle

Terminal side of the angleTerminal

side of the angle

To change a decimal degree to degrees, minutes, and seconds:

1. Multiply the decimal portion of the degree measure by 60 to find the number of minutes

2. Multiply the decimal portion of the minute measure by 60 to find the number of seconds.

Example:• Convert 329.125 to degrees, minutes, and seconds

To change from degrees, minutes, and seconds to a decimal:1. Multiply the minutes by (2. Multiply the seconds by (3. Add the degree, answer from #1, and answer from #2 together.

Example:Convert 3512’7” to a decimal.

Each rotation on the coordinate plane is 360. If an angle is 3.7 rotations counterclockwise, you would multiply 360 by 3.7. The answer would be 1332. If an angle is 2.5 rotations clockwise, you would multiply by -360 because it would be a negative angle. The answer would be -900.

Ex: Give the angle measure represented by each rotation:a. 9.5 rotations clockwiseb. 6.75 rotations counterclockwise

Tow angles are coterminal if they share the same terminal side.

45 405

Your questions for finding coterminal angles will look like this:

Identify all angles coterminal with each angle. Then find one positive angle and one negative angle that are coterminal with the angle.

answer

answer

answer

a. 45b. 225

a. 86b. 294

If each angle is in standard position, determine a coterminal angle that is between 0 and 360. State the quadrant in which the terminal side lies.

a. 775 b. -1297 c. 595 d. -777

Reference angles are acute angles formed by the terminal side of an angle and the x axis. They ARE ALWAYS BETWEEN 0 AND 90!!!

Copy yellow box on page 280 for reference angle rule.

Quadrant II

Quadrant III Quadrant IV

Quadrant 1

Find the measure of the reference angle for each angle.

1. 35

2. 105

3. 345

4. 225

Section 2: Trig Ratios in Right Triangles

The student will be able to find the values of trigonometric ratios for acute angles of a right triangle.

In a right triangle, one of the angles measures 90, and the remaining angles are acute and complementary. The longest side of a right triangle is the hypotenuse and it is always opposite the right angle. The other sides are either adjacent or opposite.

B C

A

b

a

c B is the right angle of this triangle. Side b is the hypotenuse. A and C are acute angles. Side a is opposite angle A and adjacent to angle C. Side c is adjacent to angle A and opposite angle C.

Copy the Trigonometric Ratios box on page 285.

Examples:

Find the values of the sin, cos, and tan for <B. 18

A

33

BC

Find the values of the sin, cos, and tan for <A.

Find the values of the sin, cos, and tan for <A.

AB

C

15

8

Find the values of the sin, cos, and tan for <C.

Copy the purple and white box on page 287.

Section 3: Trig Ratios on the Unit CircleStudents will be able to:• find the values of trig ratios on the unit circle• Find the vlues of trig functions of an angle in standard position given a point on its

terminal side.

A unit circle is a circle of radius 1 and it’s center is at the origin of a coordinate plane.

WHAT TO KNOW ABOUT THE UNIT CIRCLE:

• The radius is ALWAYS 1!!! • The angle we find is always located at the origin.

Examples:Use the unit circle to find each value:

1. Cos (180)

2. Sin (-90)

3. tan (90)

Use the unit circle to find the values of sin, cos, and tan for a 210 angle.

1. Find the reference angle for a 210 angle.2. Find the sin, cos, and tan for that reference angle. (box from yesterday’s lesson on

page 287).3. Change the sign of these trig ratios to reflect what quadrant you are in.

Find the values of the trig functions for an angle in standard position if a point with the coordinates (5, -12) lies on its terminal side.

Find the values of the trig functions for an angle in standard position if a point with the coordinates (-15, 20) lies on its terminal side.

Suppose is an angle in standard position whose terminal side lies in Quadrant IV. If tan is , find sin and cos.

Suppose is an angle in standard position whose terminal side lies in Quadrant III. If s is , find tan and cos.

Sec 4: Applying Trig RatiosThe student will be able to use trigonometry to find measures of the sides of right triangles.

If and r = 14, find q. q p

r

The circus has arrived and the roustabouts must put up the main tent in a field near town. A tab is located on the side of the tent 40 feet above the ground. A rope is tied to the tent at this point and then the rope is placed around a stake on the ground. If the angle that the rope makes with the ground is , how long is the rope? What is the distance between the bottom of the tent and the stake?

If J = , and j=12, find r.G

JR50

The chair lift at a ski resort rises at an angle of 20.75and attains a vertical height of 1200 feet. a. How far does the chair lift travel up the side of the mountain?b. A film crew in a helicopter records an overhead view of a skier’s downhill run from

where she gets off the chair lift at the top to where she gets back on the chair lift for her next run. If the helicopter follows a level flight path, what is the length of that path?

The angle of elevation is the angle between a horizontal line and the line of sight from an observer to an object at a higher level.

The angle of depression is the angle between a horizontal line and the line of sight from the observer to an object at a lower level.

The angle of elevation to the top of a tree is 30 from a point 28 feet away from the foot of the tree. Find the height of the tree rounded to the nearest feet.

A man on the deck of a ship is 15 ft above sea level. He observes that the angle of elevation of the top of a cliff is 70° and the angle of depression of its base at sea level is 50°. Find the height of the cliff and its distance from the ship.

From the top of a spire of height 50 ft, the angles of depression of two cars on a straight road at the same level as that of the base of the spire and on the same side of it are 25° and 40°. Calculate the the distance between the two cars.

A regular hexagon is inscribed in a circle with diameter 26.6 cm. Find the apothem of the hexagon.• An apothem is the measure of a line segment from the center of the polygon to the

midpoint of one of its sides. • A hexagon is a six-sided figure.

Sec 5: Solving Right Triangles

The student will be able to:• Evaluate inverse trig functions• Find missing angle measurements• Solve right triangles

When you know a trig function, but you don’t know the angle, you can use inverse trig to find the angle. Inverse trig is found on your calculator by pressing the 2nd key before the trig key. Trigs and their inverses “undo” each other.

Example:

𝑠𝑖𝑛𝑥=√22

𝑐𝑜𝑠𝑥=√32 𝑡𝑎𝑛𝑥=√3

3

7E

D

F

Find angle E.

) cos(arcos

Many cities place restrictions on the height and placement of skyscrapers in order to protect residents from completely shaded streets. If a 100 foot building casts an 88 foot shadow, what is the angle of elevation of the sun?

K

J L

Solve each triangle described, given the triangle below. Round to the nearest tenth. (Find all the missing parts)

a. K = 40, k = 26

b. j = 65, l = 55

Sec 6: Law of Sines

The student will be able to:a. Solve triangles by using the law of sines. If the measures of two angles and a side

are given.b. Find the area of a triangle if the measures of two sides and the included angle or the

measures of two angles and a side are given.

Law of Sines: 𝑎sin 𝐴

=𝑏sin𝐵

=𝑐

sin𝐶

*** We use this when we know two angles and a nonincluded side.

*** Note: these are NOT right triangles!

Examples:

Solve LMN if ,

Area of triangles that aren’t right triangles:

𝐾=12𝑏𝑐𝑠𝑖𝑛𝐴 C B

Examples:

Solve ABC if m A = 70°, c = 26, a = 25∠

Find the area of ABC if b = 21.2, c = 16.5, and A = 25

Find the area of ABC if a = 6.8, b = 9 and C = 5.

A regular octagon is inscribed in a circle with radius of 5 feet. Find the area of the octagon.

Sec 8: Law of Cosines:

The student will be able to:a. Solve triangles by using the law of cosinesb. Find the area of triangles if the measures of the three sides are given.

Law of Cosines:

𝑎2=𝑏2+𝑐2−2𝑏𝑐 cos 𝐴𝑏2=𝑎2+𝑐2−2𝑏𝑐 cos 𝐵𝑐2=𝑎2+𝑏2−2𝑏𝑐 cos𝐶

Solve the triangle:

A = 39.4

a = 19, b = 24.3, c = 21.8

Suppose you want to fence a triangular lot as shown below. If two sides measure 84 feet and 78 feet and the angle between the two sides is 102, what is the length of the fence to the nearest foot?

Find the area of the triangle ABC if a = 24, b = 52, and c = 39.