Triangle Trigonometry

Grade 11 · geometry · 97 practice problems · read aloud

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🔺 Triangle Trigonometry: SOH CAH TOA

Triangle Trigonometry uses the ratios between sides of right-angled triangles to find missing angles and side lengths. It's essential for solving real-world problems in fields like engineering, physics, and architecture.

Step-by-Step Guide

  1. Identify the right angle and label the sides relative to your target angle: Opposite, Adjacent, and Hypotenuse.
  2. Choose the correct ratio: Sine (sin), Cosine (cos), or Tangent (tan).
  3. Set up the equation using SOH CAH TOA.
  4. Solve for the unknown variable using algebra and a calculator.

Visual Examples

Example 1: Finding a Side

Triangle: Angle = 35°, Hypotenuse = 10, find the Opposite side.

  1. SOH: sin(θ) = Opposite / Hypotenuse
  2. sin(35°) = O / 10
  3. O = 10 × sin(35°)
  4. O ≈ 10 × 0.5736 ≈ 5.74

Example 2: Finding an Angle

Triangle: Opposite = 7, Adjacent = 5, find angle θ.

  1. TOA: tan(θ) = Opposite / Adjacent
  2. tan(θ) = 7 / 5 = 1.4
  3. θ = tan⁻¹(1.4)
  4. θ ≈ 54.5°

⚠️ Common Mistakes

  • Mislabeling sides: The hypotenuse is always opposite the right angle. Double-check which side is adjacent to your angle.
  • Calculator mode: Ensure your calculator is in DEGREE mode, not radians, when working with angles.
  • Incorrect inverse function: Use sin⁻¹ to find an angle from a ratio, not sin.

💡 Tips & Tricks

  • SOH CAH TOA: This memory aid is your best friend! (Sine = Opposite/Hypotenuse, etc.)
  • Label First: Always label the triangle's sides before choosing a ratio.
  • Check Reasonableness: The hypotenuse is always the longest side. If your calculation says otherwise, re-check your work.

Practice Suggestions

  • Start with simple problems where you identify which ratio to use.
  • Draw your own right triangles with random measurements and solve for all missing sides and angles.
  • Use online platforms like Khan Academy for randomized practice problems.
  • Create flashcards for the SOH CAH TOA ratios and common angle values (30°, 45°, 60°).

Practice problems

6 of the 97, worked through step by step — try them before opening the answer.

1 sin(2π/3) = ?

Hint: Consider the unit circle and reference angles to evaluate trigonometric functions of special angles.

Show the answer

Answer: √3/2

  1. Understand the problem. We are asked to find the value of sin(2π/3). The angle is given in radians.
  2. Convert radians to degrees for easier understanding (optional but helpful). We know that π radians = 180 degrees. So, 2π/3 radians = (2/3) * 180 degrees = 120 degrees. Therefore, sin(2π/3) = sin(120°).
  3. Locate the angle on the unit circle. An angle of 120° is in the second quadrant (between 90° and 180°).
  4. Find the reference angle. The reference angle is the acute angle the terminal side makes with the x-axis. For an angle in the second quadrant, the reference angle is 180° - θ. So, the reference angle for 120° is 180° - 120° = 60°.
  5. Recall the sine of the reference angle. We know that sin(60°) = √3 / 2.
  6. Determine the sign of sine in the second quadrant. In the second quadrant, sine is positive. Therefore, sin(120°) is positive.
  7. Combine the sign and the value. Since sin(120°) is positive and its magnitude is the same as sin(60°), we have: sin(120°) = + sin(60°) = √3 / 2.
  8. State the final answer. Since sin(2π/3) = sin(120°), the final answer is √3 / 2.

2 2sin(75°)cos(75°) = ?

Hint: This expression matches a double-angle identity pattern. Consider how sin(2θ) relates to sinθ and cosθ.

Show the answer

Answer: 0.5

  1. Recognize the double-angle identity: sin(2θ) = 2sinθcosθ
  2. Apply the identity: 2sin(75°)cos(75°) = sin(2 × 75°)
  3. Calculate the angle: 2 × 75° = 150°
  4. Evaluate sin(150°) = 0.5
  5. Therefore, 2sin(75°)cos(75°) = 0.5

The answer is 0.5.

3 2sin(15°)cos(15°) = ?

Hint: This expression matches a double-angle identity pattern. Consider how sin(2θ) relates to sinθ and cosθ.

Show the answer

Answer: 0.5

  1. Recognize that 2sinθcosθ = sin(2θ) using the double-angle identity
  2. Substitute θ = 15° into the identity: 2sin(15°)cos(15°) = sin(2 × 15°)
  3. Calculate 2 × 15° = 30°
  4. Evaluate sin(30°) = 1/2 = 0.5
  5. Therefore, 2sin(15°)cos(15°) = 0.5

The answer is 0.5.

4 sin(π/12)cos(π/12) = ?

Hint: Consider using a double-angle identity to simplify the product of sine and cosine functions.

Show the answer

Answer: 1/4

  1. Use the double-angle identity: sin(2θ) = 2sin(θ)cos(θ)
  2. Rearrange to get: sin(θ)cos(θ) = sin(2θ)/2
  3. Apply this identity with θ = π/12
  4. sin(π/12)cos(π/12) = sin(2 × π/12)/2 = sin(π/6)/2
  5. sin(π/6) = 1/2
  6. Therefore, sin(π/12)cos(π/12) = (1/2)/2 = 1/4

The answer is 1/4.

5 sin(π/3) + cos(π/6) = ?

Hint: Recall the exact values of trigonometric functions for standard angles like 30°, 45°, and 60°.

Show the answer

Answer: √3

  1. Recall the value of sin(π/3) We know π/3 radians is 60 degrees. The sine of 60 degrees is √3/2. So, sin(π/3) = √3/2.
  2. Recall the value of cos(π/6) We know π/6 radians is 30 degrees. The cosine of 30 degrees is √3/2. So, cos(π/6) = √3/2.
  3. Add the two values sin(π/3) + cos(π/6) = √3/2 + √3/2.
  4. Combine the terms Both terms have the same denominator, so we add the numerators: (√3 + √3)/2 = (2√3)/2.
  5. Simplify (2√3)/2 = √3. Final Answer: √3

6 sin(2x) = 2sin(x)cos(x) for x = 45°

  1. A) true
  2. B) false

Hint: This involves verifying a trigonometric identity at a specific angle. Consider using known values of sine and cosine for common angles to check if both sides are equal.

Show the answer

Answer: A) true

  1. Write down the left-hand side (LHS) of the equation. LHS = sin(2x) = sin(2 * 45°) = sin(90°)
  2. Recall the value of sin(90°). sin(90°) = 1 So LHS = 1.
  3. Write down the right-hand side (RHS) of the equation. RHS = 2sin(x)cos(x) = 2sin(45°)cos(45°)
  4. Recall the values of sin(45°) and cos(45°). sin(45°) = √2 / 2 cos(45°) = √2 / 2
  5. Substitute these values into the RHS. RHS = 2 * (√2 / 2) * (√2 / 2)
  6. Simplify the expression. First, 2 * (√2 / 2) = √2 So RHS = √2 * (√2 / 2) = (√2 * √2) / 2
  7. Simplify further. √2 * √2 = 2 So RHS = 2 / 2 = 1
  8. Compare LHS and RHS. LHS = 1 RHS = 1 Since LHS = RHS, the equation is true. Therefore, the statement is correct.

Let's check if sin(2x) = 2sin(x)cos(x) for x = 45°.

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