3.5 The Trig Functions

sine

cosine

cosecantsiny

r

secant

tangent cotangent

cosx

r

tany

x

cscr

y

secr

x

cotx

y

sine and cosine are only 2 of the trig functions!Here are all 6!

, x ≠ 0

, x ≠ 0

, y ≠ 0

, y ≠ 0

Ex 1) The terminal side of an angle θ in standard position passes through (–1, 7). Draw the reference triangle and evaluate the six trig functions of θ.

5 2 r–1

7 r

θ

72 + (–1)2 = r2 50 = r2

7 2 7 2sin

105 2 2

y

r

1 2 2cos

105 2 2

x

r

7tan 7

1

y

x

5 2csc

7

r

y

5 2sec 5 2

1

r

x

1cot

7

x

y

r always (+)

Ex 2) Determine the value of secθ if cosθ = 0.11

1 1sec 9.09

cos 0.11

1csc

sin

1sec

cos

1

cottan

A relationship among the 6 trig functions is they can pair up & make pairs of reciprocal functions. (as always den ≠ 0)

If we know the value of one trig function & the quadrant of θ, we can get the other 5

Ex 3) Angle in standard position, Quadrant IV and 2 2 2

2

2

7 ( 6)

49 36

13

13

x

x

x

x

–67

13cos

7

6sin

7

θx

6 6 13tan

1313

7 7 13sec

1313

13cot

6

7csc

6

13

Ex 4) Suppose that cos θ = 0.42 and

Use the symmetry of the unit circle to find the exact values of the following.

a) cos(–θ)

02

θ–θ

x-value is the sameso cos(–θ) = 0.42

b) cos(θ + π)

θ

+ π

x-value is negativeso cos(θ + π) = –0.42

c) cos(θ + 2π)

θ

+ 2π

right where you started so cos(θ + 2π) = 0.42

Homework

#305 Pg 150 #1, 5, 9, 13, 15, 17, 21, 26, 27, 29, 31, 33, 37, 43, 44, 45, 46