Section 5.3 Trigonometric Functions on the

Unit Circle

Standard position

Unit Circle

Given a unit circle, the radius or hypotenuse = 1

Sine and Cosine

MEMORIZE THIS:The right angle is always on the x axis.The acute angle is always at the origin.

MEMORIZE THIS TOO:The ordered pair of the point where the terminal side of the angle intersects the circle is (x, y) where cosine Ѳ = x and sine Ѳ = y.

45° – 45° – 90° Triangle

s

s

2s

30° – 60° – 90° Triangle

MEMORIZE THE TRIG RATIOS FOR THE SPECIAL RIGHT TRIANGLES IN

THE FIRST QUADRANT

These ratios are repeated in each quadrantaround the circle, with sign changes.

Signs of Trigonometric Functions in each Quadrant

+ - - - - + + - + -

+ - - - - + + - + -

Signs of the trig functions in the 4 quadrants

• Quadrant I = All (are positive)• Quadrant II = Students (Sine & Cosecant are +)• Quadrant I = Take (Tangent & Cotangent are +) • Quadrant I = Calculus (Cosine & Secant are +)

AS

T C

Use the unit circle to find the value for the six trigonometric functions for a

135° angle.

Tan 210°

Sin 315°

Cos 225°

Consider an angle with a point on its terminating side of (5, -12). That would be in the 4th quadrant.

Find the values of the six trigonometric functions for an angle Ѳ in standard position if a point with the coordinates (-15,20) lies on its terminal side.

If you know the value of one of the trig. functions and the quadrant in which the terminal side of Ѳ lies, you can find the values of the remaining 5 functions.

Suppose Ѳ is an angle in standard position whose terminal side lies in Quadrant IV. If

Find the values of the five remaining functions

of Ѳ.

529sec

Now try these on page 296 #1-13:

1. Why is csc 1800 undefined?2. Show that the value of sin Ѳ increases from 00 to

900 and decreases from 900 to 1800 .3. Confirm that 4. Complete the chart for the signs of the trig

functions in each quadrant.

coscotsin

Function Quadrant. I Quadrant. II Quadrant. III Quadrant. IV

Sin α & Cos αCos α & Sec α

Tan α & Cos α

Use your unit circle to find the exact measure for each of the following.

1. Tan 1800

2. Sec -900

3. Tan 450

4. Cot 2700

5. Tan 1350

6. Csc 2700

7. Tan 3600

8. Sec 1800

Find the values of the six trig functions for an angle θ in standard position if a point with the given coordinates lies on its terminal side.

1. (3,5)

2. (-6,6)

Use the unit circle to find the sin (-90°)

Use the unit circle to find the cot (270°)

Use the unit circle to find the value for the six trigonometric functions for a

210° angle.