Warm up Solve for the missing side length

Essential Question: How to right triangles relate to the unit circle? How can I use special triangles to find coordinates on the unit circle?

Standard: MM4A2. Students will use the circle to define the trigonometric functions.

c. Find values of trigonometric functions using points on the terminal sides of angles in the standard position.

e. Find values of trigonometric functions using the unit circle.

Math IV Lesson 22

Review of the Pythagorean theorem

Using the Pythagorean theorem

9

The trigonometric functions are

sine, cosine, tangent, cotangent, secant, and cosecant.

opp

adj

hyp

θ

sin = cos = tan =

csc = sec = cot = opp

hypadjhyp

hypadj

adjopp

oppadj

hypopp

Note: sine and cosecant are reciprocals, cosine and secant are reciprocals and tangent and cotangent are reciprocals

The Trigonometric Functionson the unit circle (we learned these last Thursday)

Let t be a real number and let (x,y) be the point on the unit circle corresponding to t

Sin(t) = y csc(t) = 1/y

Cos(t) = x sec(t) = 1/x

Tan(t) = y/x cot(t) = x/y

A Unit Circle has Radians, degrees and coordinates

A circle defined by x2 + y2 = 1

(x, y)

12

The Unit Circle

• Imagine the real number line wrapped around the circle.

• Each real number t corresponds to a point (x, y).

• Since the radius is 1, the number t would correspond with the central angle (s = rθ).

(1, 0)

(0, -1)

(-1, 0)

(0, 1)

t

θ= t

13

Geometry of the 45-45-90 triangle

Consider an isosceles right triangle with a hypotenuse the length of 1.

22

22

45

451 What would be the length of the sides?

2245cos

2245sin oo

14

21

23

Geometry of the 30-60-90 triangle

1

30

60○

2160cos

2360sin

2330cos

2130sin

oo

oo

Consider a 30-60-90 triangle with a hypotenuse the length of 1.

What would be the length of the sides?

Signs on the unit circle

Use special right triangles to fill in the coordinates on the unit circle

Evaluating trigonometric functions using special triangles

Solve each triangle. Redraw the triangles here and write in the lengths of the sides. a. 120 degrees b. 135 degrees c. 150 degrees

Evaluate each function without using a calculator. (Draw special right triangles in position on the Unit Circle and apply the Unit Circle Definition

of the trigonometric functions.)

• 1. Sin(240 degrees) • 2. cos(315 degrees)

Wednesday

• Quiz on the coordinates of the unit circle!!!!