Evaluating Trig Functions Of Any Angle

TUTORIAL

Evaluating Trig Functions Of Any Angle

TUTORIALClick the speaker icon

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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!