This document contains solutions to trigonometric equations worked out in multiple steps. It finds the solutions to the equation 8sin^2θ - 5sinθ - 1 = 0 in the interval 0 ≤ θ < 2π. The solutions are found to be θ ≈ {.90, 2.24, 3.30, 6.12} by using trigonometric identities and properties of sinusoidal functions.

1. 7.5 Trigonometric Equations
Day Two
Isaiah 33:2
O LORD, be gracious to us; we long for you. Be our strength
every morning, our salvation in time of distress.

2. 8. Groups: Find all solutions of:
2
2sin θ − sin θ − 1 = 0

3. 8. Groups: Find all solutions of:
2
2sin θ − sin θ − 1 = 0
using Quadratic Formula program:
1
sin θ = 1 sin θ = −
2
π 7π 11π
θ= θ= ,
2 6 6
π 7π 11π
θ = + 2kπ , + 2kπ , + 2kπ
2 6 6

4. 9. Groups: Solve graphically for −π ≤ x ≤ π
1.05sin x − 1.07 = 0

5. 9. Groups: Solve graphically for −π ≤ x ≤ π
1.05sin x − 1.07 = 0
∅

6. 9. Groups: Solve graphically for −π ≤ x ≤ π
1.05sin x − 1.07 = 0
∅
Discuss how this could have been
obtained without graphing.

7. 10. Groups: Find all solutions in 0 ≤ θ < 2π
2
8sin θ − 5sin θ − 1 = 0

8. 10. Groups: Find all solutions in 0 ≤ θ < 2π
2
8sin θ − 5sin θ − 1 = 0
sin θ ≈ .7844 sin θ ≈ −.1594

9. 10. Groups: Find all solutions in 0 ≤ θ < 2π
2
8sin θ − 5sin θ − 1 = 0
sin θ ≈ .7844 sin θ ≈ −.1594
θ ≈ .90 QI

10. 10. Groups: Find all solutions in 0 ≤ θ < 2π
2
8sin θ − 5sin θ − 1 = 0
sin θ ≈ .7844 sin θ ≈ −.1594
θ ≈ .90 QI
θ ≈ π − .90

11. 10. Groups: Find all solutions in 0 ≤ θ < 2π
2
8sin θ − 5sin θ − 1 = 0
sin θ ≈ .7844 sin θ ≈ −.1594
θ ≈ .90 QI
θ ≈ π − .90
θ ≈ 2.24 QII

12. 10. Groups: Find all solutions in 0 ≤ θ < 2π
2
8sin θ − 5sin θ − 1 = 0
sin θ ≈ .7844 sin θ ≈ −.1594
θ ≈ .90 QI θ ≈ −.16
θ ≈ π − .90
θ ≈ 2.24 QII

13. 10. Groups: Find all solutions in 0 ≤ θ < 2π
2
8sin θ − 5sin θ − 1 = 0
sin θ ≈ .7844 sin θ ≈ −.1594
θ ≈ .90 QI θ ≈ −.16
θ ≈ π − .90 ∴ θ ≈ 2π − .16
θ ≈ 2.24 QII

14. 10. Groups: Find all solutions in 0 ≤ θ < 2π
2
8sin θ − 5sin θ − 1 = 0
sin θ ≈ .7844 sin θ ≈ −.1594
θ ≈ .90 QI θ ≈ −.16
θ ≈ π − .90 ∴ θ ≈ 2π − .16
θ ≈ 2.24 QII θ ≈ 6.12 QIV

15. 10. Groups: Find all solutions in 0 ≤ θ < 2π
2
8sin θ − 5sin θ − 1 = 0
sin θ ≈ .7844 sin θ ≈ −.1594
θ ≈ .90 QI θ ≈ −.16
θ ≈ π − .90 ∴ θ ≈ 2π − .16
θ ≈ 2.24 QII θ ≈ 6.12 QIV
θ ≈ π + .16

16. 10. Groups: Find all solutions in 0 ≤ θ < 2π
2
8sin θ − 5sin θ − 1 = 0
sin θ ≈ .7844 sin θ ≈ −.1594
θ ≈ .90 QI θ ≈ −.16
θ ≈ π − .90 ∴ θ ≈ 2π − .16
θ ≈ 2.24 QII θ ≈ 6.12 QIV
θ ≈ π + .16
θ ≈ 3.30 QIII

17. 10. Groups: Find all solutions in 0 ≤ θ < 2π
2
8sin θ − 5sin θ − 1 = 0
sin θ ≈ .7844 sin θ ≈ −.1594
θ ≈ .90 QI θ ≈ −.16
θ ≈ π − .90 ∴ θ ≈ 2π − .16
θ ≈ 2.24 QII θ ≈ 6.12 QIV
θ ≈ π + .16
θ ≈ 3.30 QIII
{.90, 2.24, 3.30, 6.12}

18. HW #10
Quiz Tomorrow!!
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in the joy of achievement, in the thrill of creative
effort.
Franklin D. Roosevelt