PreCalc 2 · Analytic Trigonometry
Find an exact value of sin 2θ, cos 2θ, or tan 2θ from what you know about θ.
Sine
sin 2θ = 2 sin θ cos θ
Cosine, three ways
cos 2θ = cos² θ − sin² θ
cos 2θ = 2 cos² θ − 1
cos 2θ = 1 − 2 sin² θ
Tangent
tan 2θ = 2 tan θ1 − tan² θ
A double-angle expression
2 sin θ cos θ is sin 2θ.
cos² θ − sin² θ is cos 2θ.
Replace the expression with the single angle, then evaluate.
You are given both ratios, a single tangent, or a double-angle expression.
sin 2θ = 2 sin θ cos θ
sin θ = 45
cos θ = 35
2 · 45 · 35
2425
Answer: 2425
sin 2θ = 2 sin θ cos θ
sin θ = 35
cos θ = 45
2 · 35 · 45
2425
Answer: 2425
cos 2θ = cos² θ − sin² θ
cos θ = 45
cos² θ = 1625
sin θ = 35
sin² θ = 925
1625 − 925
725
Answer: 725
sin 2θ = 2 sin θ cos θ
sin θ = 513
cos θ = 1213
2 · 513 · 1213
120169
Answer: 120169
cos 2θ = cos² θ − sin² θ
cos θ = 1213
cos² θ = 144169
sin θ = 513
sin² θ = 25169
144169 − 25169
119169
Answer: 119169
tan 2θ = 2 tan θ1 − tan² θ
tan θ = 12
2 tan θ
1
tan² θ = 14
1 − tan² θ
1 − 14
44 − 14
34
134
1 · 43
43
Answer: 43
tan 2θ = 2 tan θ1 − tan² θ
tan θ = 3
2 tan θ
6
tan² θ = 9
1 − tan² θ
1 − 9
−8
6−8
6 · −18
−34
Answer: −34
sin 2θ = 2 sin θ cos θ
2 sin 45° cos 45° is sin 90°.
sin 90° = 1
Answer: 1
cos 2θ = cos² θ − sin² θ
cos² 30° − sin² 30° is cos 60°.
cos 60° = 12
Answer: 12
sin 2θ = 2 sin θ cos θ
sin θ = 725
cos θ = 2425
2 · 725 · 2425
336625
Answer: 336625
tan 2θ = 2 tan θ1 − tan² θ
tan θ = 512
2 tan θ
56
tan² θ = 25144
1 − tan² θ
1 − 25144
144144 − 25144
119144
56119144
56 · 144119
120119
Answer: 120119
You are given one ratio and a quadrant. Find the other ratio before you substitute.
sin 2θ = 2 sin θ cos θ
sin² θ + cos² θ = 1
cos² θ = 1 − sin² θ
sin θ = 45
sin² θ = 1625
1 − 1625
2525 − 1625
925
cos² θ = 925
θ is in Quadrant II, so cosine is negative.
cos θ = −35
sin θ = 45
cos θ = −35
2 · 45 · (−35)
−2425
Answer: −2425
sin 2θ = 2 sin θ cos θ
sin² θ + cos² θ = 1
cos² θ = 1 − sin² θ
sin θ = 35
sin² θ = 925
1 − 925
2525 − 925
1625
cos² θ = 1625
θ is in Quadrant II, so cosine is negative.
cos θ = −45
sin θ = 35
cos θ = −45
2 · 35 · (−45)
−2425
Answer: −2425
sin 2θ = 2 sin θ cos θ
sin² θ + cos² θ = 1
cos² θ = 1 − sin² θ
sin θ = 513
sin² θ = 25169
1 − 25169
169169 − 25169
144169
cos² θ = 144169
θ is in Quadrant II, so cosine is negative.
cos θ = −1213
sin θ = 513
cos θ = −1213
2 · 513 · (−1213)
−120169
Answer: −120169
sin 2θ = 2 sin θ cos θ
sin² θ + cos² θ = 1
sin² θ = 1 − cos² θ
cos θ = 817
cos² θ = 64289
1 − 64289
289289 − 64289
225289
sin² θ = 225289
θ is in Quadrant IV, so sine is negative.
sin θ = −1517
sin θ = −1517
cos θ = 817
2 · (−1517) · 817
−240289
Answer: −240289
sin 2θ = 2 sin θ cos θ
sin² θ + cos² θ = 1
sin² θ = 1 − cos² θ
cos θ = −1517
cos² θ = 225289
1 − 225289
289289 − 225289
64289
sin² θ = 64289
θ is in Quadrant III, so sine is negative.
sin θ = −817
sin θ = −817
cos θ = −1517
2 · (−817) · (−1517)
240289
Answer: 240289
sin 2θ = 2 sin θ cos θ
sin² θ + cos² θ = 1
cos² θ = 1 − sin² θ
sin θ = −725
sin² θ = 49625
1 − 49625
625625 − 49625
576625
cos² θ = 576625
θ is in Quadrant III, so cosine is negative.
cos θ = −2425
sin θ = −725
cos θ = −2425
2 · (−725) · (−2425)
336625
Answer: 336625
sin 2θ = 2 sin θ cos θ
sin² θ + cos² θ = 1
cos² θ = 1 − sin² θ
sin θ = −817
sin² θ = 64289
1 − 64289
289289 − 64289
225289
cos² θ = 225289
θ is in Quadrant IV, so cosine is positive.
cos θ = 1517
sin θ = −817
cos θ = 1517
2 · (−817) · 1517
−240289
Answer: −240289
tan 2θ = 2 tan θ1 − tan² θ
sin² θ + cos² θ = 1
cos² θ = 1 − sin² θ
sin θ = 1213
sin² θ = 144169
1 − 144169
169169 − 144169
25169
cos² θ = 25169
θ is in Quadrant I, so cosine is positive.
cos θ = 513
tan θ = 125
2 tan θ
245
tan² θ = 14425
1 − tan² θ
1 − 14425
2525 − 14425
−11925
245−11925
245 · −25119
−120119
Answer: −120119
tan 2θ = 2 tan θ1 − tan² θ
tan θ = −512
2 tan θ
−56
tan² θ = 25144
1 − tan² θ
1 − 25144
144144 − 25144
119144
−56119144
−56 · 144119
−120119
Answer: −120119
cos 2θ = cos² θ − sin² θ
sin² θ + cos² θ = 1
sin² θ = 1 − cos² θ
cos θ = −35
cos² θ = 925
1 − 925
2525 − 925
1625
sin² θ = 1625
θ is in Quadrant II, so sine is positive.
sin θ = 45
cos θ = −35
cos² θ = 925
sin θ = 45
sin² θ = 1625
925 − 1625
−725
Answer: −725
tan 2θ = 2 tan θ1 − tan² θ
tan θ = 25
2 tan θ
45
tan² θ = 425
1 − tan² θ
1 − 425
2525 − 425
2125
452125
45 · 2521
2021
Answer: 2021
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