Here are the key steps to solve this problem: 1) Use the half-angle formula for cosine: cos(2α) = 1 - 2sin^2(α) 2) Given: sec(α) = 2 Use the identity: sec^2(α) = 1 + tan^2(α) Solve for tan(α): tan(α) = -√3 3) Substitute tan(α) = -√3 into the half-angle formula for sine: sin(α) = ±√(1-cos^2(α)/2) sin(α) = ±1/2 4) Given that tan(α) < 0

1. 7.3 Double Angle, Half Angle, and
Product Sum Formulas
On your help sheet ...
Psalm 119:28
My soul is weary with sorrow; strengthen me according to
your word.

2. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ

3. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ

4. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
−5
−12 13

5. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
2
cos 2θ = 1− 2sin θ
−5
−12 13

6. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
2
cos 2θ = 1− 2sin θ
−5 2
⎛ 12 ⎞
= 1− 2 ⎜ − ⎟
−12 13 ⎝ 13 ⎠

7. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
2
cos 2θ = 1− 2sin θ
−5 2
⎛ 12 ⎞
= 1− 2 ⎜ − ⎟
−12 13 ⎝ 13 ⎠
144
= 1− 2 ⋅
169

8. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
2
cos 2θ = 1− 2sin θ
−5 2
⎛ 12 ⎞
= 1− 2 ⎜ − ⎟
−12 13 ⎝ 13 ⎠
144
= 1− 2 ⋅
169
169 288
= −
169 169

9. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
2
cos 2θ = 1− 2sin θ
−5 2
⎛ 12 ⎞
= 1− 2 ⎜ − ⎟
−12 13 ⎝ 13 ⎠
144
= 1− 2 ⋅
169
169 288
= −
169 169
119
=−
169

10. Do this with your group:
12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of sin 2θ
−5
−12 13

11. Do this with your group:
12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of sin 2θ
−5 ⎛ 12 ⎞ ⎛ 5 ⎞
sin 2θ = 2 ⎜ − ⎟ ⎜ − ⎟
⎝ 13 ⎠ ⎝ 13 ⎠
−12 13
120
=
169

12. Do this with your group:
12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of tan 2θ
−5
−12 13

13. Do this with your group:
12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of tan 2θ
⎛ −12 ⎞
−5 2 ⎜ ⎟
⎝ −5 ⎠
tan 2θ = 2
−12 13 ⎛ −12 ⎞
1− ⎜ ⎟
⎝ −5 ⎠
24 24
120
= 5 = 5 =−
144 119 119
1− −
25 25

14. Prove the identity:
cos 2x + 1 2
4
= 1+ tan x
2 cos x

15. Prove the identity:
cos 2x + 1 2
4
= 1+ tan x
2 cos x
2
2 cos x − 1+ 1
4
=
2 cos x

16. Prove the identity:
cos 2x + 1 2
4
= 1+ tan x
2 cos x
2
2 cos x − 1+ 1
4
=
2 cos x
2
2 cos x
4
=
2 cos x

17. Prove the identity:
cos 2x + 1 2
4
= 1+ tan x
2 cos x
2
2 cos x − 1+ 1
4
=
2 cos x
2
2 cos x
4
=
2 cos x
1
2
=
cos x

18. Prove the identity:
cos 2x + 1 2
4
= 1+ tan x
2 cos x
2
2 cos x − 1+ 1
4
=
2 cos x
2
2 cos x
4
=
2 cos x
1
2
=
cos x
2
1+ tan x =

19. The formulas for Lowering Powers are used to
derive the Half Angle formulas.
I’ve put these Lowering Powers formulas on your
help sheet so you can see them and have them, but
we won’t be doing any problems which require them.

20. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2

21. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2

22. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
2

23. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
− 3
2

24. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2

25. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2
1
1+
=± 2
2

26. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2
1
1+
=± 2
2
3
=±
4

27. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2
1
1+
=± 2
2
3
=±
4
3
=±
2

28. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2
1
1+
=± 2
2
Which?
3 + or - ??
=±
4
3
=±
2

29. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2 α is in QIV
1
1+
=± 2
2
Which?
3 + or - ??
=±
4
3
=±
2

30. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2 α is in QIV
1 3π
1+ ∴ < α < 2π
2 2
=±
2
Which?
3 + or - ??
=±
4
3
=±
2

31. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2 α is in QIV
1 3π
1+ ∴ < α < 2π
2 2
=±
2 3π α
Which? and < <π
3 + or - ?? 4 2
=±
4
3
=±
2

32. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2 α is in QIV
1 3π
1+ ∴ < α < 2π
2 2
=±
2 3π α
Which? and < <π
3 + or - ?? 4 2
=±
4 this is in QII
3 α
=± so cos is negative
2 2

33. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2 α is in QIV
1 3π
1+ ∴ < α < 2π
2 2
=±
2 3π α
Which? and < <π
3 + or - ?? 4 2
=±
4 this is in QII
3 α
=± so cos is negative
2 2
3
=−
2

34. Do this with your group:
α
If sec α = 2 and tan α < 0 , find the exact value of sin
2
1
− 3
2

35. Do this with your group:
α
If sec α = 2 and tan α < 0 , find the exact value of sin
2
1
1 α 1−
sin = 2
− 3 2 2
2
1
=
4
1
=
2

36. Do this with your group:
α
If sec α = 2 and tan α < 0 , find the exact value of tan
2
1
− 3
2

37. Do this with your group:
α
If sec α = 2 and tan α < 0 , find the exact value of tan
2
3
1 α −
tan = 2
− 3 2 1+ 1
2
2
3
−
= 2
3
2
3
=−
3

38. HW #5
Optimism is the faith that leads to achievement.
Nothing can be done without hope and confidence.
Helen Keller