THE UNIT CIRCLE Precalculus Trigonometric Functions

Precalculus WU 10/14 Find one positive and one negative angle co-terminal with the given angle.

Objectives: Find trig function values for special angles using the unit circle. Evaluate Trig functions using the unit circle. Use domain and period to evaluate trig functions. Solve application problems using the unit circle.

Trigonometric Functions The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. The sides of the right triangle are: θ hyp  the side opposite the acute angle , opp  the side adjacent to the acute angle , adj  and the hypotenuse of the right triangle. The trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. sin  = cos  = tan  = csc  = sec  = cot  = opp hyp adj Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Trigonometric Functions

Definition: Trigonometric Functions of Any Angle Let  be an angle in standard position with (x, y), a point on the terminal side of  and r = y x r (x, y)  Definition: Trigonometric Functions of Any Angle

Example: Trigonometric Functions of Any Angle Determine the exact values of the six trigonometric functions of the angle . y x (3, 6)  Example: Trigonometric Functions of Any Angle

So, we know that trigonometric function values are side length relationships of right triangles. We can easily evaluate the exact values of trigonometric functions for special angles.

Geometry of the 30-60-90 Triangle 2 Consider an equilateral triangle with each side of length 2. 60○ 30○ 30○ The three sides are equal, so the angles are equal; each is 60. The perpendicular bisector of the base bisects the opposite angle. 1 1 Use the Pythagorean Theorem to find the length of the altitude, . Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Geometry of the 30-60-90 Triangle

Special right triangle relationships

Now, let’s apply it to the unit circle… What does “unit circle” really mean? It’s a circle with a radius of 1 unit. What is the equation of the “unit circle”?

Let’s begin with an easy family… What are the coordinates? Now, reflect the triangle to the second quadrant…

What are the coordinates? Now, reflect the triangle to the third quadrant…

What are the coordinates? Now, reflect the triangle to the fourth quadrant…

What are the coordinates?

Now, reflect the triangle to the second quadrant. Complete the family… . Now, reflect the triangle to the second quadrant.

Now, reflect the triangle to the third quadrant.

Now, reflect the triangle to the fourth quadrant. What are the coordinates? Now, reflect the triangle to the fourth quadrant.

What are the coordinates?

Let’s look at another “family” Now, reflect the triangle to the second quadrant

What are the coordinates? Now, reflect the triangle to the third quadrant

What are the coordinates? Now, reflect the triangle to the fourth quadrant

What are the coordinates?

Trigonometric Values of Common Angles Ordered pairs of special angles around the Unit Circle x y (0, 1) 90° 120° 60° Since r = 1… 135° 45° 30° 150° (–1, 0) 180° (1, 0) 0° 360° 210° 330° 315° 225° 240° 300° (0, –1) 270° Trigonometric Values of Common Angles

Important point: Since r = 1… Because ordered pairs around the unit circle (x, y) represent sine and cosine, and the equation of the circle is , We have the following identity:

What if the radius is not 1? 6 1 Trigonometric values are functions of the angle – ratios of sides of similar triangles remain the same. So it always holds that .

Trigonometric Values of Common Angles Trigonometric Values of Special Angles x y (0, 1) 90° 120° 60° 135° 45° 30° 150° (–1, 0) 180° (1, 0) 0° 360° 210° 330° 315° 225° 240° 300° (0, –1) 270° Trigonometric Values of Common Angles

Domain and Range of Sine and Cosine The domain of the sine and cosine function is the set of all real numbers. (1, 0) (–1, 0) (0, –1) (0, 1) x y Unit Circle Range The point (x, y) is on the unit circle, therefore the range of the sine and cosine function is between – 1 and 1 inclusive. Domain and Range of Sine and Cosine

Definition: Periodic Functions A function f is periodic if there is 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. x y Unit Circle Periodic Function t = 0, 2, … Period Definition: Periodic Functions

Example: Periodic Functions Evaluate sin 5 using its period. 5 - 2 - 2 =  sin 5 = sin  = 0 x y (–1, 0) Adding 2 to each value of t in the interval [0, 2] completes another revolution around the unit circle. Example: Periodic Functions

You Try: Evaluate sin Evaluate cos

Can you? Evaluate each of the following. Exact values only please.

Even and Odd Trig Functions (1, 0) (–1, 0) (0,–1) (0,1) x y Remember: if f(-t) = f(t) the function is even if f(-t) = - f(t) the function is odd The cosine and secant functions are EVEN. cos(-t)=cos t sec(-t)=sec t The sine, cosecant, tangent, and cotangent functions are ODD. sin(-t)= -sin t csc(-t)= -csc t tan(-t)= -tan t cot(-t)= -cot t

Evaluating Trigonometric Functions Example: Evaluate the six trigonometric functions at  = . (1, 0) (–1, 0) (0, –1) (0, 1) x y  Evaluating Trigonometric Functions

Evaluate the six trigonometric functions at Example: Evaluate the six trigonometric functions at y = x =

Example continued: -2

Evaluate the six trigonometric functions at You Try: Evaluate the six trigonometric functions at (0, -1) = -1 = y = -1 = x = 0 DNE = 0 DNE Does Not Exist

So, you think you got it now? You Try: Evaluate the six trigonometric functions at -1 Sin Cos Tan Csc Sec Cot -1 So, you think you got it now?

Application: A ladder 20 feet long leans against the side of a house. The angle of elevation of the ladder is 60 degrees. Find the height from the top of the ladder to the ground.

Application: An airplane flies at an altitude of 6 miles toward a point directly over an observer. If the angle of elevation from the observer to the plane is 45 degrees, find the horizontal distance between the observer and the plane. .

Homework 4.2 pg. 264 1-51 odd

Trig Races

HWQ 10/15 Evaluate each of the following. Exact values only please.