Loading...

About Trigonometric Equations & Identities — Class 10 IB

Prove basic trigonometric identities and solve trigonometric equations within given domains. This topic is part of the IB Class 10 mathematics syllabus (chapter: Unit 13). On this page you can practice 40 questions across three difficulty levels — 10 easy, 20 medium, and 10 hard — each with a visual step-by-step solution, plus a timed 27-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Trigonometric Equations & Identities — solved examples for Class 10 IB

Example 1easy

Ravi was asked to prove the identity (sin θ + cos θ)² = 1 + 2 sin θ cos θ. He wrote the following steps:
1. (sin θ + cos θ)² = sin² θ + cos² θ
2. sin² θ + cos² θ = 1
3. Therefore, (sin θ + cos θ)² = 1.
Which of the following statements correctly identifies Ravi's mistake?
  1. A)A. In Step 1, Ravi incorrectly expanded the expression (sin θ + cos θ)².
  2. B)B. In Step 2, Ravi incorrectly applied the Pythagorean identity.
  3. C)C. Ravi started with the wrong side of the equation to prove the identity.
  4. D)D. There is no mistake; Ravi's proof is correct.

Step-by-step solution

  1. The algebraic identity for squaring a binomial is (a + b)² = a² + 2ab + b².
  2. In Step 1, Ravi expanded (sin θ + cos θ)² as sin² θ + cos² θ, which is incorrect.
  3. The correct expansion should be (sin θ + cos θ)² = sin² θ + cos² θ + 2 sin θ cos θ.
  4. After this correction, using sin² θ + cos² θ = 1, the expression becomes 1 + 2 sin θ cos θ, thus proving the identity.

Answer: A. In Step 1, Ravi incorrectly expanded the expression (sin θ + cos θ)².

Example 2medium

Which of the following expressions is equivalent to (1 + sinθ)(1 - sinθ)?
  1. A)1
  2. B)sin²θ
  3. C)cos²θ
  4. D)2cosθ

Step-by-step solution

  1. The expression is in the form (a + b)(a - b), which simplifies to a² - b².
  2. So, (1 + sinθ)(1 - sinθ) = 1² - sin²θ = 1 - sin²θ.
  3. Using the fundamental trigonometric identity sin²θ + cos²θ = 1, we can rearrange it to 1 - sin²θ = cos²θ.

Answer: cos²θ

Example 3hard

Find the sum of all solutions for the equation `2cos²θ + 3sinθ = 0` in the interval `0° ≤ θ < 360°`.
  1. A)180°
  2. B)270°
  3. C)360°
  4. D)540°

Step-by-step solution

  1. Substitute `cos²θ = 1 - sin²θ` into the equation: `2(1 - sin²θ) + 3sinθ = 0`.
  2. Rearrange to form a quadratic equation: `2 - 2sin²θ + 3sinθ = 0` which becomes `2sin²θ - 3sinθ - 2 = 0`.
  3. Factor the quadratic: `(2sinθ + 1)(sinθ - 2) = 0`. This gives `sinθ = -1/2` or `sinθ = 2` (the latter has no solution).
  4. For `sinθ = -1/2`, the principal angle is `30°`. In `0° ≤ θ < 360°`, the solutions are `θ = 180° + 30° = 210°` and `θ = 360° - 30° = 330°`. The sum of these solutions is `210° + 330° = 540°`.

Answer: 540°

Practice questions on Trigonometric Equations & Identities

  1. Q1.easy

    Which statement is true about the trigonometric equation sin x = 2?
    1. A)A. It has an infinite number of solutions.
    2. B)B. It has exactly two solutions in the interval 0° ≤ x < 360°.
    3. C)C. It has no real solutions.
    4. D)D. It has a solution at x = 90°.
    Show answer

    Answer: C. It has no real solutions.

    Hint: Consider the range of the sine function.

  2. Q2.easy

    Which of the following expressions is equivalent to (sec²θ - 1)?
    1. A)A. cosec²θ
    2. B)B. sin²θ
    3. C)C. tan²θ
    4. D)D. cos²θ
    Show answer

    Answer: C. tan²θ

    Hint: Recall the Pythagorean identity involving secant and tangent.

  3. Q3.easy

    What is the smallest positive solution for the equation tan x = -1?
    1. A)A. 45°
    2. B)B. 135°
    3. C)C. 225°
    4. D)D. 315°
    Show answer

    Answer: B. 135°

    Hint: First, find the reference angle for tan x = 1. Then, consider the quadrants where tan x is negative.

  4. Q4.medium

    A student is proving the identity (secθ - tanθ)(secθ + tanθ) = 1. Which of the following is the most appropriate first step in their proof?
    1. A)Convert secθ to 1/cosθ and tanθ to sinθ/cosθ
    2. B)Expand the left side using the difference of squares formula
    3. C)Square both sides of the equation
    4. D)Assume the identity is true and work backwards
    Show answer

    Answer: Expand the left side using the difference of squares formula

    Hint: Look for algebraic patterns first. The expression on the left side resembles a common algebraic identity.

  5. Q5.medium

    If tanθ = 5/12 and θ is an acute angle, what is the value of (sinθ + cosθ)?
    1. A)17/13
    2. B)7/13
    3. C)13/12
    4. D)12/5
    Show answer

    Answer: 17/13

    Hint: Draw a right-angled triangle or use the identity 1 + tan²θ = sec²θ to find the hypotenuse or other ratios.

  6. Q6.medium

    Ravi was solving the equation 2sin²x - 1 = 0 for 0° ≤ x < 360°. His steps are:
    1. 2sin²x = 1
    2. sin²x = 1/2
    3. sinx = 1/√2
    4. x = 45°, 135°
    What mistake, if any, did Ravi make?
    1. A)No mistake, the solution is correct.
    2. B)In step 3, he forgot the negative square root.
    3. C)In step 4, he only considered solutions in the first and second quadrants.
    4. D)He should have used the identity cos(2x) = 1 - 2sin²x.
    Show answer

    Answer: In step 3, he forgot the negative square root.

    Hint: When you take the square root of both sides of an equation, remember that there are usually two possible values: positive and negative.

  7. Q7.hard

    Which of the following expressions is equivalent to `(1 + sinθ) / cosθ + cosθ / (1 + sinθ)`?
    1. A)2tanθ
    2. B)2secθ
    3. C)2cosecθ
    4. D)2cotθ
    Show answer

    Answer: 2secθ

    Hint: Combine the fractions by finding a common denominator and then simplify the numerator using identities.

  8. Q8.hard

    Find the smallest positive solution for the equation `3tan²θ - 2√3 tanθ - 3 = 0`.
    1. A)30°
    2. B)60°
    3. C)120°
    4. D)150°
    Show answer

    Answer: 120°

    Hint: Treat this as a quadratic equation in `tanθ`. You can factor it or use the quadratic formula to find the values of `tanθ`.

  9. Q9.hard

    A student attempts to prove the identity `(cosecθ - sinθ)(secθ - cosθ) = 1 / (tanθ + cotθ)`. They follow these steps:
    Step 1: Convert LHS to `(1/sinθ - sinθ)(1/cosθ - cosθ)`
    Step 2: Simplify LHS to `((1 - sin²θ)/sinθ)((1 - cos²θ)/cosθ)`
    Step 3: Further simplify LHS to `(cos²θ/sinθ)(sin²θ/cosθ)`
    Step 4: Cancel terms to get `cosθ sinθ`
    Step 5: Convert RHS to `1 / (sinθ/cosθ + cosθ/sinθ)`
    Step 6: Simplify RHS to `1 / ((sin²θ + cos²θ)/(sinθcosθ))`
    Step 7: Simplify RHS to `1 / (1/(sinθcosθ))`
    Step 8: Final RHS is `sinθcosθ`.
    Step 9: Conclude LHS = RHS, so the identity is proven.

    Which of the following statements is true about the student's proof?
    1. A)The student made an error in Step 3.
    2. B)The student made an error in Step 5.
    3. C)The student made an error in Step 7.
    4. D)The proof is entirely correct.
    Show answer

    Answer: The proof is entirely correct.

    Hint: Carefully review each step of the student's work. Focus on algebraic manipulation and the application of trigonometric identities.

These are 9 of the 40 questions available for Trigonometric Equations & Identities. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.