Trigonometry: The Unit Circle - WordPress.com · 2016-01-08 · Honors Precalculus Trigonometry: The Unit Circle and Trig Functions . The Unit Circle •A unit circle is a circle

BELLWORK

Assuming the Earth makes one complete revolution around its axis in exactly 24 hours, and assuming the radius of the Earth is 3,960 miles:

a) What central angle, in radians, does the Earth rotate in one second?

b) … In degrees?

c) What distance through space, in feet, would a person standing still on the surface of the Earth actually travel in that time?

d) How long would it take for a person, sitting still on the surface of the Earth, to travel a distance of 4.321 miles through space?

REMEMBER:

𝜔 =𝜃

𝑡

𝑠 = 𝑟𝜃

1 revolution = 2𝜋 rads = 360°

Mr. Velazquez

Honors Precalculus

Trigonometry: The Unit Circle and

Trig Functions

The Unit Circle

• A unit circle is a circle of radius 1, centered at the origin.

• If we draw a central angle in standard position on the unit circle, the radian measure of the central angle will be equal to the arc length created by that angle.

The Six Trigonometric Functions

If 𝜃 is the radian measure of the central angle in a unit circle, and 𝑃(𝑥, 𝑦) is a point on the unit circle that corresponds to 𝜃, then we define the trig functions in the following way:

1

𝜃

(1, 0)

(0, 1)

(–1, 0)

(0, –1)

𝑃(𝑥, 𝑦)

𝑦

𝑥

𝐬𝐢𝐧 𝜽 = 𝒚 𝐜𝐬𝐜 𝜽 =𝟏

𝒚

𝐜𝐨𝐬 𝜽 = 𝒙

𝐭𝐚𝐧 𝜽 =𝒚

𝒙

𝐬𝐞𝐜 𝜽 =𝟏

𝒙

𝐜𝐨𝐭 𝜽 =𝒚

𝒙

Basic Unit Circle Angles

Draw this circle on a piece of paper (rough sketch; no perfection necessary), and follow along!

Positive and Negative Trig Functions

Find the value of each trigonometric function

cot sec4 4

Basic Trig Identities

Basic Trig Identities

5 2 5Given sin t = and cos t= find the value of

5 5

each of the four remaining trigonometric functions.

Basic Trig Identities

1 3Given sin t = and cos t= find the value of

2 2

each of the four remaining trigonometric functions.

Basic Trig Identities

10Given that cos t= and 0 t< , find the

10 2

value of sin t using a trigonometric identity.

Periodic Functions

Periodic Functions

Periodic Functions

Find the value of each trigonometric function

9a. cot

4

5b. cos

2

3c. sec -

4

Evaluating With a Calculator

Use a calculator to find the value to four decimal places:

9a. cot

4

5b. cos

2

3c. sec -

4

d. csc 1.2

Exit Ticket/Homework

Assuming: sin 𝜃 = 𝑎, cos 𝜃 = 𝑏, and tan 𝜃 = 𝑐

Rewrite the following expressions in terms of 𝑎, 𝑏, and 𝑐:

1. sin(−𝜃) − sin 𝜃 2. 3 cos(−𝜃) − cos 𝜃

3. sin(𝜃 + 2𝜋) − cos 𝜃 + 4𝜋 + tan 𝜃 + 𝜋

4. sin(−𝜃 + 100𝜋) + cos(−𝜃) + cos(𝜃 + 50𝜋)

Homework Assignment: Pg. 487, #20-60 (mult of 2)