Pre-Calc Book Section 5.2Trigonometric Functions

of Real Numbers

Objectives:• How do you find the value of any given trig angle?• How can understanding the signs of the quadrants help in finding the function value of any trig angle?

Trigonometric Functions of Any Angle

siny

r

cosx

r

tan , 0yx

x

csc , 0ry

y

sec , 0rx

x

cot , 0xy

y

Ex. 1 Find the six trig functions of each given real number t.

a)

b)

6t

2t

Ex.2 Let be a point on the terminal side of . Find the sine, cosine, and tangent of .

1 3,2 2

Ex.3 Let (-3,4) be a point on the terminal side of . Find the sine, cosine, and tangent of .

Signs of Trig Functions

Quad PositiveIIIIIIIV

, ,

, ,

All Students Take Calculus

All Trig. Functions

Sine/Cosecant

Tangent/Cotangent

Cosine/Secant

Ex. 4 Determine the sign of the following trig functions.

a)

b)

c)

cos3

tan 4

7sin

4

> 0, because the terminal point of is in Quad I3

t

> 0, because the terminal point of is in Quad III4t

< 0, because the terminal point of is in Quad IV

7

4t

Class WorkFind the six trig functions of each given real number t.

1. 2. 4

t

3

t

Determine the sign of the following trig functions.

3.

4.

5.

5tan

6

cos7

5csc

4

Ex 5. Evaluate the sine, cosine and tangent of the given angle.

) 570a

) 60b

13)4

c

11)3

d

Ex. 6 Find the exact value of the trig function at the given real number t.

a)

b)

c)

2cos

3

tan3

19sin

4

d)

e)

f)

2cos

3

5sin

3

5tan

6

Class WorkFind the exact value of the trig function at the given real number t.

6.

7.

8.

5sin

6

4tan

3

14cos

3

Ex.7 State the quadrant in which lies.

) cos 0, sin 0a

) sec 0, tan 0b

) sin 0, cot 0c

Fundamental Identites

Ex 8. If and t is in quadrant IV, find the

values of all the trig functions at t.

3cos

5t

Ex 9. Find the values of the six trig functions of in the given quadrant.

5) tan cos 0

4a

) csc 3 tan 0b

2) sin cot 0

7c

Class WorkFind the values of all the trig functions of t from the given information.

13. In quad III

14. In quad III

15. sint < 0

4cos

5t

1tan

4t

sec 2t

HW #3 p.416 3-19 eoo, 27-35 eoo, 63-69 odd