Evaluating Trig Functions Of Any Angle TUTORIAL
Post on 15-Jan-2016
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Evaluating Trig Functions Of Any Angle
TUTORIAL
Evaluating Trig Functions Of Any Angle
TUTORIALClick the speaker icon
on each slide to hear the narration
First Concept: Evaluating a trig function of a special angle
First Concept: Evaluating a trig function of a special angle
1. Sketch the angle in standard position
2. Determine the reference angle
3. Draw the triangle showing x, y, r with their values, based on the side ratios of the reference angle
4. Take the appropriate ratio of the sides
5. Simplify your ratio, if necessary
1. Sketch the angle in standard position
2. Determine the reference angle
3. Draw the triangle showing x, y, r with their values, based on the side ratios of the reference angle
4. Take the appropriate ratio of the sides
5. Simplify your ratio, if necessary
Example: Example:
Sketch the terminal side of the angle in standard position
Sketch the terminal side of the angle in standard position
tan5π3
5π3
Example: Example:
Find the reference angle: Find the reference angle:
tan5π3
π3
= 60°
Example: Example:
Label the values of x, y, and r, paying close attention to the signs (r is always positive):
Label the values of x, y, and r, paying close attention to the signs (r is always positive):
tan5π3
60
1
2− 3
Example: Example:
Compute the tangent ratio: Compute the tangent ratio:
tan5π3
60
1
2− 3
tan5π3
=yx
=− 3−1
= 3
Second Concept: Trig functions of angles that lie on the axes
Second Concept: Trig functions of angles that lie on the axes
§ Trig functions of 90, 180, 270, and 360 can be tricky
§ Steps:
§ Draw the angle and indicate x, y, and r
§ Use the following definitions:
§ Trig functions of 90, 180, 270, and 360 can be tricky
§ Steps:
§ Draw the angle and indicate x, y, and r
§ Use the following definitions:
sinθ =yr
cosθ =xr
tanθ =yx
Example: Functions of 270Example: Functions of 270
Draw the angle in standard position Draw the angle in standard position
270
Example: Functions of 270Example: Functions of 270
Indicate the coordinates of the endpoint of the terminal ray (always make r = 1)
Indicate the coordinates of the endpoint of the terminal ray (always make r = 1)
x = 0y = –1r = 1
Example: Functions of 270Example: Functions of 270
Take the appropriate ratios to compute sin, cos, and tan
Take the appropriate ratios to compute sin, cos, and tan
x = 0y = –1r = 1
sin 270° =yr
=−11
=−1
cos270° =xr
=01
=0
tan270° =yx
=−10
=undefined
Third Concept: Angles with similar ratios
Third Concept: Angles with similar ratios
§ Every angle with the same reference angle will have a similar ratio
§ Identical to each other, or…§ Different sign from each other
§ Use knowledge of the quadrants and x, y, r to know whether the ratio is positive or negative in that quadrant
§ r is always positive
§ Every angle with the same reference angle will have a similar ratio
§ Identical to each other, or…§ Different sign from each other
§ Use knowledge of the quadrants and x, y, r to know whether the ratio is positive or negative in that quadrant
§ r is always positive
sinθ =yr
cosθ =xr
tanθ =yx
Signs of functions in each quadrant
Signs of functions in each quadrant
sinθ = y/r, so sin is positive where y is positive (Quadrants 1 and 2) and negative where y is negative (Quadrants 3 and 4)
cosθ = x/r, so cos is positive where x is positive (Quadrants 1 and 4) and negative where x is negative (Quadrants 2 and 3)
tanθ = y/x, so tan is positive where x and y have the same sign (Quadrants 1 and 3) and negative where x and y have different signs (Quadrants 2 and 4)
sinθ = y/r, so sin is positive where y is positive (Quadrants 1 and 2) and negative where y is negative (Quadrants 3 and 4)
cosθ = x/r, so cos is positive where x is positive (Quadrants 1 and 4) and negative where x is negative (Quadrants 2 and 3)
tanθ = y/x, so tan is positive where x and y have the same sign (Quadrants 1 and 3) and negative where x and y have different signs (Quadrants 2 and 4)
Summary chartSummary chart
All trig
functions are positive
Sin is
positive, others are negative
Tan is
positive, others are negative
Cos is
positive, others are negative
x and y are positive
x is negy is pos
x and y are negative
x is posy is neg AS
T C
“All Students Take Calculus”“All Schools Torture Children”
“Avoid Silly Trig Classes”
Mnemonic:
Example: cos 35 = 0.819Example: cos 35 = 0.819
Other angles in the family (meaning they have a reference angle equal to 35) In the second quadrant, 145 has the same reference
angle and the cosine is negative, socos 145 = –0.819
In the third quadrant, 215 has the same reference angle and the cosine is negative, socos 215 = –0.819
In the fourth quadrant, 325 has the same reference angle and the cosine is positive, so cos 325 = 0.819
Other angles in the family (meaning they have a reference angle equal to 35) In the second quadrant, 145 has the same reference
angle and the cosine is negative, socos 145 = –0.819
In the third quadrant, 215 has the same reference angle and the cosine is negative, socos 215 = –0.819
In the fourth quadrant, 325 has the same reference angle and the cosine is positive, so cos 325 = 0.819
The EndThe End
Hope you enjoyed the show! Hope you enjoyed the show!
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