Section 2.3
Periodic Properties
cos(θ + 2π) = cos θ
sin(θ + 2π) = sin θ
tan(θ + π) = tan θ
sec(θ + 2π) = sec θ
csc(θ + 2π) = csc θ
cot(θ + π) = cot θ
Use the fact that the trigonometric functions are periodic to find the exact value of each
expression.
1.
sin 405◦
2.
cos(−420◦ )
3.
sec 390◦
4.
cot 540◦
5.
tan 690◦
!
6.
25π
tan
4
!
7.
19π
sin
6
8.
13π
cos
3
9.
31π
sin
4
10.
cos(31π)
!
!
Section 2.3
Let P = (x, y) be a point on the unit circle that corresponds to the angle θ. If we know in which
quadrant the point P lies, then we can determine the signs of the trigonometric functions of θ
Q2
Q1
x<0
y>0
x>0
y>0
cos(θ) < 0
sin(θ) > 0
cos(θ) > 0
sin(θ) > 0
Q3
Q4
x<0
y<0
x>0
y<0
cos(θ) < 0
sin(θ) < 0
cos(θ) > 0
sin(θ) < 0
For the following questions, name the quadrant in which the angle lies in.
11.
sin θ < 0 and cos θ < 0
11.
12.
sin θ < 0 and tan θ < 0
12.
13.
csc θ > 0 and cos θ < 0
13.
14.
sec θ < 0 and tan θ > 0
14.
In the next set of problems, the values for cos θ and sin θ are given. Find the exact value
of each of the four remaining trigonometric functions. (Note: rationalize all denominators,
when appropriate.)
√
2 2
1
and sin θ = −
15. cos θ =
3
3
√
√
5
2 5
16. cos θ =
and sin θ =
5
5
FUNDAMENTAL IDENTITIES
Reciprocal Identities
csc(θ) =
1
sin(θ)
Quotient Identities
tan(θ) =
sec(θ) =
sin(θ)
cos(θ)
1
cos(θ)
cot(θ) =
cot(θ) =
1
tan(θ)
1
tan(θ)
Pythagorean Identities
sin2 (θ) + cos2 (θ) = 1
tan2 (θ) + 1 = sec2 (θ)
1 + cot2 (θ) = csc2 (θ)
2
n
We write sin2 (θ) when we mean sin(θ) . In general, we write sinn (θ) instead of sin(θ) whenever n is an integer not equal to −1. This convention applies to the other five trig functions.
Derive the pythagorean identities from the definitions of sine and cosine.
Find the exact value of each of the remaining trig functions of θ.
17.
18.
19.
2
3
1
tan θ = −
3
4
cot θ =
3
sin θ = −
π<θ<
3π
2
sin θ > 0
cos θ < 0
Even and Odd Properties
Sine, cosecant, tangent and cotangent are odd functions. Cosine and secant are even functions.
sin(−θ) = − sin(θ)
cos(−θ) = cos(θ)
tan(−θ) = − tan(θ)
csc(−θ) = − csc(θ)
sec(−θ) = sec(θ)
cot(−θ) = − cot(θ)
We will prove these identities later on once we have established the difference of angles formula.
Use the even and odd properties to find the exact values of each expression.
20.
sin(−30◦ )
21.
cos(−60◦ )
22.
tan(−180◦ )