7.4 The other trigonometric functions  (Page 6/14)

Key equations

Key concepts

• The tangent of an angle is the ratio of the y -value to the x -value of the corresponding point on the unit circle.
• The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function.
• The six trigonometric functions can be found from a point on the unit circle. See [link] .
• Trigonometric functions can also be found from an angle. See [link] .
• Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See [link] .
• A function is said to be even if $f(-x)=f(x)$ and odd if $f\left(-x\right)=-f\left(x\right)$ for all x in the domain of f.
• Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
• Even and odd properties can be used to evaluate trigonometric functions. See [link] .
• The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
• Identities can be used to evaluate trigonometric functions. See [link] and [link] .
• Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See [link] .The trigonometric functions repeat at regular intervals.
• The period $P$ of a repeating function $f$ is the smallest interval such that $f(x+P)=f(x)$ for any value of $x.$
• The values of trigonometric functions can be found by mathematical analysis. See [link] and [link] .
• To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See [link] .

Section exercises

Verbal

Algebraic

For the following exercises, find the exact value of each expression.

For the following exercises, use reference angles to evaluate the expression.