Lesson 4.2 Trigonometric Functions: The Unit Circle Essential Question: How do you evaluate trigonometric functions by using the unit circle?

The Unit Circle The two historical perspectives of trigonometry incorporate different methods of introducing the trigonometric functions. Our first introduction to these functions is based on the unit circle.

The Unit Circle Consider the unit circle given by 𝑥 2 + 𝑦 2 =1. Imagine that the real number line is wrapped around this circle, with positive numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping.

As the real number line is wrapped around the unit circle, each real number t corresponds to a point 𝑥,𝑦 on the circle. The real number 0 corresponds to the point (1, 0). Moreover, because the unit circle has a circumference of 2𝜋, the real number 2𝜋 also corresponds to the point (1, 0).

In general, each real number t also corresponds to a central angle 𝜃 (in standard position) whose radian measure is t. With this interpretation of t, the arc length formula 𝑠=𝑟𝜃 (with r = 1) indicates that the real number t is the (directional) length of the arc intercepted by the angle 𝜃, given in radians.

The Trigonometric Functions The coordinates x and y are two functions of the real variable t. You can use these coordinates to define the six trigonometric functions of t. sine cosine tangent cosecant secant cotangent These six functions are normally abbreviated sin, cos, tan, csc, sec, and cot, respectively.

Definition of Trigonometric Functions Let t be a real number and let (x, y) be the point on the unit circle corresponding to t. sin 𝑡 =𝑦 cos 𝑡 =𝑥 tan 𝑡 = 𝑦 𝑥 , 𝑥≠0 csc 𝑡= 1 𝑦 , 𝑦≠0 sec 𝑡 = 1 𝑥 , 𝑥≠0 cot 𝑡 = 𝑥 𝑦 , 𝑦≠0

The Unit Circle

How do you evaluate trigonometric functions by using the unit circle? Find the corresponding point on the unit circle. Use the definition of the trigonometric function to evaluate it.

Evaluate the six trigonometric functions at each real number. 𝑡= 𝜋 6

Evaluate the six trigonometric functions at each real number. 𝑡= 5𝜋 4

Evaluate the six trigonometric functions at each real number. 𝑡=𝜋

Evaluate the six trigonometric functions at each real number. 𝑡=− 𝜋 3

Evaluate the six trigonometric functions at each real number. 𝑡= 2𝜋 3

Evaluate the six trigonometric functions at each real number. 𝑡= 4𝜋 3

Evaluate the six trigonometric functions at each real number. 𝑡= 𝜋 2

Evaluate the six trigonometric functions at each real number. 𝑡=− 2𝜋 3

Domain of Sine and Cosine The domain of the sine and cosine functions is the set of all real numbers.

Range of Sine and Cosine To determine the range of these two functions, consider the unit circle. Because r = 1, it follows that sin 𝑡=𝑦 and cos 𝑡 =𝑥. Moreover, because (x, y) is on the unit circle, you know that −1≤𝑦≤1 and −1≤𝑥≤1. So, the values of sine and cosine also range between – 1 and 1. −1≤𝑦≤1   −1≤𝑥≤1 and −1≤ sin 𝑡 ≤1 −1≤ cos 𝑡 ≤1

Period of Sine and Cosine Adding 2𝜋 to each value of t in the interval 0,2𝜋 completes a second revolution around the unit circle. The values of sin 𝑡+2𝜋 and cos 𝑡+2𝜋 correspond to those of sin 𝑡 and cos 𝑡 . Similar results can be obtained for repeated revolutions (positive or negative) around the unit circle. This leads to the general result sin 𝑡+2𝜋𝑛 = sin 𝑡 and cos 𝑡+2𝜋𝑛 = cos 𝑡 for any integer n and real number t. Functions that behave in such a repetitive (or cyclic) manner are called periodic.

Definition of Periodic Function A function f is periodic when there exists a positive real number c such that 𝑓 𝑡+𝑐 =𝑓 𝑡 for all t in the domain of f. The least number c for which f is periodic is called the period of f.

Even and Odd Functions Even Function Odd Function A function whose graph is symmetric with respect to the y-axis is an even function. Odd Function A function whose graph is symmetric with respect to the origin is an odd function.

Test for Even and Odd Functions A function f is even when, for each x in the domain of f, 𝑓 −𝑥 =𝑓 𝑥 . A function f is odd when, for each x in the domain of f, 𝑓 −𝑥 =−𝑓 𝑥 .

Even and Odd Trigonometric Functions The cosine and secant functions are even. The sine, cosecant, tangent, and cotangent functions are odd. cos −𝑡 = cos 𝑡 sec −𝑡 = sec 𝑡 sin −𝑡 =− sin 𝑡 csc −𝑡 =− csc 𝑡 tan −𝑡 =− tan 𝑡 cot −𝑡 =− cot 𝑡

Evaluate cos 9𝜋 3 .

Evaluate sin 13𝜋 6 .

Evaluate cos− 7𝜋 2 .

Evaluate sin − 11𝜋 2 .

If sin 𝑡= 4 5 , find sin −𝑡 .

If tan 𝑡= 2 3 , find tan −𝑡 .

Use a calculator to evaluate the trigonometric functions. sin 5𝜋 7

Use a calculator to evaluate the trigonometric functions. cos 2𝜋 3

Use a calculator to evaluate the trigonometric functions. cot 1.5

How do you evaluate trigonometric functions by using the unit circle?

Ticket Out the Door Using the unit circle find: sin 5𝜋 6