Download Trigonometric function identities

Trigonometric function identities
Our Favorite (capitol ‘F’) trig identities are
1. (symmetries)
sin(−θ) = − sin(θ),
and
cos(−θ) = cos(θ)
2. (pythagorian identity)
sin2 (θ) + cos2 (θ) = 1
3. (angle addition formulas)
sin(θ + φ) = sin(θ) cos(φ) + sin(φ) cos(θ),
and
cos(θ + φ) = cos(θ) cos(φ) − sin(θ) sin(φ)
Using what we know about the relation between points on the unit circle and the functions sin(θ) and cos(θ),
explain/prove the first two identities. Draw pictures.
Trigonometric function identities
For the following problems, use the three basic identities (symmetries, pythagorean, angle addition) to prove
the given equalities.
tan x + tan y
.
1 − tan x tan y
1. tan(x + y) =
14. 1 + tan2 (π/2 − x) =
1
.
cos2 (π/2 − x)
15.
sin A cos A
+
= 1.
csc A sec A
3. cos 3x = cos3 x − 3 cos x sin2 x.
16.
tan B
sec B
−
= 1.
cos B
cot B
4. sin 3x = 3 cos2 x sin x − sin3 x.
17.
1
1
sec2 w
+ sec2 w +
=2+
.
2
2
csc w
sec w
csc2 w
5. sin2 A cot2 A = (1 − sin A)(1 + sin A).
18. sec4 V − sec2 V =
r
2. sin(x/2) = ±
6. tan B =
7.
1 − cos x
.
2
cos B
.
sin B cot2 B
tan V cos V
= 1.
sin V
8. sin E cot E + cos E tan E = sin E + cos E.
9.
10.
1
1
+
.
cot4 V
cot2 V
19. sin4 x + cos2 x = cos4 x + sin2 x.
20. tan 3α =
3 tan α − tan3 α
.
1 − 3 tan2 α
21. cot(α/2) =
sin α
.
1 − cos α
√
1
1
+
− 1 = 0.
sec2 x csc2 x
22. cos(π/6 − x) + cos(π/6 + x) =
sec A − 1 cos A − 1
+
= 0.
sec A + 1 cos A + 1
23. sin(α + β) sin(α − β) = sin2 α − sin2 β.
2
24. sin(π/3 − x) + sin(π/3 + x) =
√
3 cos x.
3 cos x.
11. sin V (1 + cot V ) = csc V .
12.
sin(π/2 − w)
= cot w.
cos(π/2 − w)
25. cos(π/4 − x) − cos(π/4 + x) =
√
2 sin x.
26. 2 sin α cos β = sin(α + β) + sin(α − β).
1
13. sec(π/2 − z) =
.
sin z
27. 2 sin α sin β = cos(α − β) − cos(α + β).
More fun with trigonometric function identities
For the following problems, use the three identities above to prove the given equalities.
1. cos 2θ = 2 sin(π/4 + θ) sin(π/4 − θ).
2. (1/2) sin 2A =
3. cot(x/2) =
tan A
.
1 + tan2 A
1 + cos x
.
sin x
19.
tan 2x
.
tan x
A
A
sin − cos
2
2
2
2
11. cos4 A =
1 − sin A
= (sec A − tan A)2 .
1 + sin A
22.
tan A − sin A
sin3 A
=
.
sec A
1 + cos A
23.
2 tan2 A
= 1 − cos 2A.
1 + tan2 A
.
24. tan 2A = tan A +
2 cos 2A + cos 2A + 1
.
4
25. sin 2A =
12.
tan
sin A + sin B
=
sin A − sin B
tan
A+B
2 .
A−B
2
26.
13.
cot2 θ
.
1 + cot2 θ
21. cos B cos(A + B) + sin B sin(A + B) = cos A.
sin 2x
9. cot x =
.
1 − cos 2x
10. 1 − sin A =
sin(β − α)
.
sin α sin β
20. (tan A − cot A)2 + 4 = sec2 A + csc2 A.
8. csc A sec A = 2 csc 2A .
1 + sin 2A
cos A + sin A
=
.
cos A − sin A
cos 2A
18. cos2 θ =
2
.
1 + tan2 A
7. tan 2x tan x + 2 =
15.
17. tan θ csc θ cos θ = 1.
1 − tan2 θ
= cos 2θ.
1 + tan2 θ
6. 1 + cos 2A =
cos 2A
cot A − 1
=
.
1 + sin 2A
cot A + 1
16. cot α − cot β =
4. sin 2B(cot B + tan B) = 2.
5.
14.
tan A
.
cos 2A
2 tan A
.
1 + tan2 A
4 sin A
1 + sin A 1 − sin A
=
−
.
2
1 − sin A 1 + sin A
1 − sin A
sin α + sin 3α
= tan 2α.
cos α + cos 3α
27. tan A + sin A =
csc A + cot A
.
csc A cot A