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Questions and Answers

Solve the equation: 2cos x + sin x - 1 = 0.

x = π/3, 5π/3

Solve the equation: sin(2x) = cos x.

x = π/4, 3π/4, 5π/4, 7π/4

Solve the equation: (tan x - 1)(cos x - 1) = 0.

x = π/4, 0, π

Solve the equation: (2cos x - 3)(2sin x - 1) = 0.

x = 5π/6, 3π/2 Signup and view all the answers

Solve the equation: cos(2x) = cos x.

x = 0, π/3, 2π/3, π Signup and view all the answers

Solve the equation: 4sin^2 x + 4cos^2 x - 5 = 0.

x = π/4, 3π/4, 5π/4, 7π/4 Signup and view all the answers

Solve the equation: 2tan^2 x + 5tan x + 3 = 0.

x = π/3, -1/2 Signup and view all the answers

Solve the equation: cos x - 5 = 3cos x + 6.

x = 3π/2 Signup and view all the answers

Solve the equation using a calculator: sin x = 0.8246.

x ≈ 0.974, 2.167 Signup and view all the answers

Solve the equation using a calculator: cos x = -0.4721.

x ≈ 2.113, 3.029 Signup and view all the answers

Solve the equation using a calculator: sin x = 1.1414.

No solution Signup and view all the answers

Solve the equation using a calculator: tan x = -1.1285.

x ≈ 3.944, 5.319 Signup and view all the answers

Simplify into a single Trig function: cos(x) sec(x).

1 Signup and view all the answers

Simplify into a single Trig function: tan(x) - cot(-x).

tan(x) + cot(x) Signup and view all the answers

Simplify into a single Trig function: (sin x + cos x)^2 - 1.

2sin(x)cos(x) Signup and view all the answers

Simplify into a single Trig function: 2cos(x).

2cos(x) Signup and view all the answers

Using a double-angle or half-angle identity: sin θ = -13/25 and 270° < θ < 360°. Find cos 2θ.

cos 2θ = -24/25 Signup and view all the answers

Using a double-angle or half-angle identity: cos θ = 4/5 and 270° < θ < 360°. Find sin 2θ.

sin 2θ = -24/25 Signup and view all the answers

Using a double-angle or half-angle identity: sin θ = -3/5 and π/2 < θ < 2π. Find tan θ.

tan θ = 3/4 Signup and view all the answers

Using a double-angle or half-angle identity: cot θ = -3/91 and π/2 < θ < 2π. Find tan 2θ.

tan 2θ = -91/3 Signup and view all the answers

Using a double-angle or half-angle identity: sec θ = 13/2 − 5 and π/2 < θ < π. Find cos 2θ.

cos 2θ = -24/25 Signup and view all the answers

Find the exact value: sin(-15°).

sin(-15°) = -sin(15°) = -1/2(√6 - √2) Signup and view all the answers

Find the exact value: cos(-105°).

cos(-105°) = cos(105°) Signup and view all the answers

Find the exact value: tan(5π/12).

tan(5π/12) = (tan(3π/12) + tan(2π/12)) / (1 - tan(3π/12)tand(2π/12)) Signup and view all the answers

Find the exact value: sin(9π/2) + cos(9π/2).

sin(9π/2) + cos(9π/2) = -1 Signup and view all the answers

Find the exact value: tan 76° + tan 164°.

tan 76° + tan 164° = tan(240°) Signup and view all the answers

Find the exact value: 1 - tan 76° tan 164°.

1 - tan 76° tan 164° = 0 Signup and view all the answers

Study Notes

Trigonometric Equations

Solve trigonometric equations by:
- Isolating the trigonometric function
- Using known trigonometric identities
- Finding the solutions within the specified interval

Solving Trigonometric Equations

1. 2cos x + sin x − 1 = 0
- Use the quadratic formula to solve for cos x: cos x = (1 ± √3) / 2
- Find the solutions for x within [0, 2π): x = π / 3, 5π / 3
2. sin (2x) = cos x
- Use the double-angle identity: sin (2x) = 2sin x cos x
- Simplify to get: 2sin x cos x - cos x = 0
- Factor: cos x (2sin x - 1) = 0
- Find the solutions for x within [0, 2π): x = π / 2, 7π / 6, 11π / 6
3. (tan x − 1)(cos x − 1) = 0
- Set each factor equal to zero
- Solve for x: x = π / 4, π
4. (2cos x − 3)(2sin x − 1) = 0
- Set each factor equal to zero
- Solve for x: x = π / 6, 7π / 6
5. cos (2x) = cos x
- Use the double-angle identity: cos (2x) = 2cos² x - 1
- Simplify to get: 2cos² x - cos x - 1 = 0
- Factor: (2cos x + 1)(cos x - 1) = 0
- Find the solutions for x within [0, 2π): x = 0, 2π / 3, 4π / 3
6. 4sin² x + 4cos² x − 5 = 0
- Use the identity sin² x + cos² x = 1
- Simplify to get: 4 - 5 = 0
- This equation has no solutions
7. 2tan² x + 5tan x + 3 = 0
- Factor the equation: (2tan x + 3)(tan x + 1) = 0
- Solve for x: x = arctan(-3/2), x = 3π / 4
8. cos x - 5 = 3cos x + 6
- Combine like terms: -2cos x = 11
- Solve for x: x = arccos(-11/2). There are no solutions in the interval [0, 2π)

Solving Trigonometric Equations with a Calculator

9. sin x = 0.8246
- Use the arcsin function to find x: x = arcsin(0.8246) ≈ 0.98, 2.16
10. cos x = -0.4721
- Use the arccos function to find x: x = arccos(-0.4721) ≈ 2.07, 4.21
11. sin x = 1.1414
- The sine function has a range of -1 to 1, so there are no solutions.
12. tan x = -1.1285
- Use the arctan function to find x: x = arctan(-1.1285) ≈ 2.31, 5.46

Simplifying Trigonometric Expressions

13. tan(x) * cos(x) * sec(x)
- Use the identity: sec(x) = 1/cos(x)
- Simplify to get: tan(x)
14. (tan(x)- cot(-x)) / sec(x)
- Use the identities: cot(-x) = -cot(x), sec(x) = 1/cos(x)
- Simplify to get: sin(x) - cos(x)
15. (sin(x) + cos(x))² - 1 / 2cos(x)
- Expand the square: sin²(x) + 2sin(x)cos(x) + cos²(x) - 1 / 2cos(x)
- Use the identity: sin²(x) + cos²(x) = 1
- Simplify to get: tan(x) + 1/2
16. (tan(y) + cot(y))sin(y) / 1
- Use the identities: tan(y) = sin(y)/cos(y), cot(y) = cos(y)/sin(y)
- Simplify to get: 1/cos(y) = sec(y)
17. (csc(x) - cot(x)) / (sec(x)-1)
- Use the identities: csc(x) = 1/sin(x), cot(x) = cos(x)/sin(x), sec(x) = 1/cos(x)
- Simplify to get: cos(x) / (1 - cos(x))
18. cos(x) / (1 - sin(x)) + tan(-x)
- Use the identities: tan(-x) = -tan(x), tan(x) = sin(x)/cos(x)
- Simplify to get: 1 / (cos(x) - sin(x))

Double-Angle and Half-Angle Identities

19. sin θ = -3/25 and 270° < θ < 360°. Find cos(θ/2)
- Use the half-angle identity: cos²(θ/2) = (1 + cosθ)/2
- Find cos θ: cos θ = √(1 - sin² θ) = 4 / 5
- Substitute values: cos²(θ/2) = (1 + 4/5)/2 = 9/10
- Solve for cos(θ/2): cos(θ/2) = 3/√10
20. cos θ = 4 / 5 and 270° < θ < 360°. Find sin 2θ
- Use the double-angle identity: sin 2θ = 2sin θ cos θ
- Find sin θ: sin θ = -√(1 - cos²θ) = -3/5
- Substitute values: sin 2θ = 2(-3/5)(4/5) = -24/25
21. sin θ = -3/5 and 3π/2 < θ < 2π. Find tan(θ/2)
- Use the half-angle identity: tan(θ/2) = sin θ / (1 + cos θ)
- Find cos θ: cos θ = √(1 - sin²θ) = 4/5
- Substitute values: tan(θ/2) = (-3/5) / (1 + 4/5) = -3/9 = -1/3
22. cot θ = -91/3 and 3π/2 < θ < 2π. Find tan 2θ
- Use the double-angle identity: tan 2θ = 2tan θ / (1 - tan² θ)
- Find tan θ: tan θ = 1/cot θ = -3/91
- Substitute values: tan 2θ = 2(-3/91) / (1 - (-3/91)²) = -57/40
23. sec θ = -5/13 and π/2 < θ < π. Find cos 2θ
- Use the double-angle identity: cos 2θ = 1 - 2sin² θ
- Find sin θ: sin θ = 1 / csc θ = 1 / (1/sec θ) = 1 / (-5/13) = -13/5
- Substitute values: cos 2θ = 1 - 2(-13/5)² = -169/25

Sum and Difference Identities

24. sin(-15°)
- Use the angle subtraction identity: sin (A - B) = sin A cos B - cos A sin B
- Rewrite -15° as 45° - 60°
- Substitute values: sin(-15°) = sin 45° cos 60° - cos 45° sin 60° = (√2/2)(1/2) - (√2/2)(√3/2) = (√2 - √6) / 4
25. cos(-105°)
- Use the angle subtraction identity: cos (A - B) = cos A cos B + sin A sin B
- Rewrite -105° as 45° - 150°
- Substitute values: cos(-105°) = cos 45° cos 150° + sin 45° sin 150° = (√2/2)(-√3/2) + (√2/2)(1/2) = (√2 - √6) / 4
26. tan(5π/12)
- Use the angle addition identity: tan (A + B) = (tan A + tan B) / (1 - tan A tan B)
- Rewrite 5π/12 as π/4 + π/6
- Substitute the values of tan (π/4) and tan (π/ 6) and simplify
27. sin (π/9) cos (2π/9) + cos (π/9) sin (2π/9)
- Use the angle addition identity: sin (A + B) = sin A cos B + cos A sin B
- Rewrite the expression as sin (π/9 + 2π/9)
- Simplify to sin (π/3)
28. (tan 76° + tan 164°) / (1 - tan 76° tan 164°)
- Use the angle addition identity: tan (A + B) = (tan A + tan B) / (1 - tan A tan B)
- Rewrite the expression as tan (76° + 164°)
- Simplify to get tan (240°) = √3