Untitled Quiz
27 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

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.

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

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

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

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

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

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

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

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

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

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

<p>x ≈ 0.974, 2.167</p> Signup and view all the answers

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

<p>x ≈ 2.113, 3.029</p> Signup and view all the answers

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

<p>No solution</p> Signup and view all the answers

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

<p>x ≈ 3.944, 5.319</p> Signup and view all the answers

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

<p>1</p> Signup and view all the answers

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

<p>tan(x) + cot(x)</p> Signup and view all the answers

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

<p>2sin(x)cos(x)</p> Signup and view all the answers

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

<p>2cos(x)</p> 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.

<p>cos 2θ = -24/25</p> 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.

<p>sin 2θ = -24/25</p> 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.

<p>tan θ = 3/4</p> 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.

<p>tan 2θ = -91/3</p> 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.

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

Find the exact value: sin(-15°).

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

Find the exact value: cos(-105°).

<p>cos(-105°) = cos(105°)</p> Signup and view all the answers

Find the exact value: tan(5π/12).

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

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

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

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

<p>tan 76° + tan 164° = tan(240°)</p> Signup and view all the answers

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

<p>1 - tan 76° tan 164° = 0</p> 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

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Quiz Team

Related Documents

Lab 6.5 Trigonometric Equations PDF

More Like This

Untitled Quiz
37 questions

Untitled Quiz

WellReceivedSquirrel7948 avatar
WellReceivedSquirrel7948
Untitled Quiz
19 questions

Untitled Quiz

TalentedFantasy1640 avatar
TalentedFantasy1640
Untitled Quiz
55 questions

Untitled Quiz

StatuesquePrimrose avatar
StatuesquePrimrose
Untitled Quiz
18 questions

Untitled Quiz

RighteousIguana avatar
RighteousIguana
Use Quizgecko on...
Browser
Open
Browser
Browser