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Trigonometric Functions of General Angles

- Unit-5 Chapter-13 Section-3
- William Marinos

What You Will Learn

- Find values of trigonometric functions for

general angles - Use reference angles to find values of

trigonometric functions

Vocab Review

- Initial side A fixed ray along the positive

x-axis - Terminal side A ray that rotates around the

center - Standard position When an angles is positioned

so that its vertex is at the origin and its

initial side is along the positive x-axis

Key Concept

- Let be an angle in standard position and

let P(x,y) be a point on the terminal side of

. Using the Pythagorean Theorem, the distance r

from the origin to P is given by r

. The trigonometric functions of an angle in

standard position may be defined as follows

- Sin
- Cos
- Tan
- Csc
- Sec
- Cot

Examples 1 (pg. 717)

- Find the exact values of the six trigonometric

functions of if the terminal side of

contains the point (5,12). - From the coordinates give, you know that x 5

and y -12. Use the Pythagorean Theorem to find

r. - Pythagorean theorem
- Replace x with 5 and y with

-12. - or 13
- Now use x 5, y -12, and r 13 to write the

ratios.

Quadrantal Angle

- If the terminal side of angle lies on one of

the axes, - is called a quadrantal angle. The quadrantal

angles are 0, 90, 180, and 270 degrees. Either X

or Y will equal 0. Since division by zero is

undefined, two of the trigonometric values are

undefined for each quadrantal angle.

Example 2

- Find the values of the six trigonometric

functions for an angle in standard position that

measures 270 degrees - When 270, x 0 and y -r

Reference Angles

- To find the values of trig functions of angles

greater than 90 degrees (or less than 0 degrees),

you need to know how to find the measures of

reference angles. If is a nonquadrantal

angle in standard position, its reference angle,

, is defined as the acute angle formed by

the terminal side of and the x-axis - The following rule can be used to find the

reference angle for any nonquadrantal angle

where 0 degrees lt lt 360 degrees (or 0 lt

lt2

THE END

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