How many practice questions are available for Trigonometry?

SparkEd Math offers 60+ practice questions for Trigonometry across 3 difficulty levels (Easy, Medium, Hard) plus a timed test mode. Each question comes with visual step-by-step solutions and AI coaching hints.

Is Trigonometry content free on SparkEd Math?

Yes! All Trigonometry content on SparkEd Math is 100% free, including practice questions, visual solutions, lessons, and AI coaching. No credit card required.

How does SparkEd Math teach Trigonometry differently from textbooks?

SparkEd Math uses visual step-by-step solutions for every Trigonometry problem, making complex concepts easy to understand. The platform includes 3 difficulty levels for progressive learning, a Super Power Help AI coach for instant doubt-clearing, and interactive practice rather than passive reading.

Is SparkEd's Trigonometry content aligned with the Olympiad syllabus?

Yes, all Trigonometry questions and lessons on SparkEd Math are fully aligned with the Mathematical Olympiad Preparation (Olympiad) syllabus for Class 10. The content follows the official curriculum structure and exam pattern.

Trigonometry — Class 10 Olympiad (2025-26)

About Trigonometry — Class 10 Olympiad

Apply trigonometric ratios, prove identities, and solve heights and distances problems at competition level. This topic is part of the Olympiad Class 10 mathematics syllabus (chapter: Module 8). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Trigonometry — solved examples for Class 10 Olympiad

Example 1easy

Evaluate the expression: (sin²1° + sin²2° + ... + sin²89°) - (cos²1° + cos²2° + ... + cos²89°).

A)0
B)1
C)44
D)89/2

Step-by-step solution

The first sum is S1 = sin²1° + sin²2° + ... + sin²89°. We can pair terms using sin²A + sin²(90°-A) = sin²A + cos²A = 1.
S1 = (sin²1° + sin²89°) + (sin²2° + sin²88°) + ... + (sin²44° + sin²46°) + sin²45°. This gives 44 pairs, each summing to 1, plus sin²45°.
So, S1 = 44 × 1 + (1/√2)² = 44 + 1/2 = 89/2.
Similarly, the second sum is S2 = cos²1° + cos²2° + ... + cos²89°. Using cos²A + cos²(90°-A) = cos²A + sin²A = 1, S2 = 89/2.
Therefore, the expression is S1 - S2 = 89/2 - 89/2 = 0.

Answer: 0

Example 2medium

A square ABCD has side length 's'. Point E is on CD such that CE = s/3. Point F is on BC such that BF = s/4. What is the value of tan(∠AEF)?

A)15/19
B)25/19
C)20/19
D)30/19

Step-by-step solution

Place D at the origin (0,0). Then A=(0,s), B=(s,s), C=(s,0).
E is on CD with CE = s/3, so E = (s - s/3, 0) = (2s/3, 0). F is on BC with BF = s/4, so F = (s, s - s/4) = (s, 3s/4).
Slope of AE (m₁): m₁ = (0 - s) / (2s/3 - 0) = -s / (2s/3) = -3/2.
Slope of EF (m₂): m₂ = (3s/4 - 0) / (s - 2s/3) = (3s/4) / (s/3) = 9/4.
tan(∠AEF) = |(m₂ - m₁) / (1 + m₁m₂)| = |(9/4 - (-3/2)) / (1 + (-3/2)(9/4))| = |(9/4 + 6/4) / (1 - 27/8)| = |(15/4) / (-19/8)| = |-30/19| = 30/19.
$tan(θ) = |(m2 - m1) / (1 + m1 * m2)|$

Answer: 30/19

Example 3hard

If sin⁴θ/a + cos⁴θ/b = 1/(a+b), then find the value of sin⁸θ/a³ + cos⁸θ/b³.

A)1/(a³+b³)
B)a/(a+b)⁴ + b/(a+b)⁴
C)1/(a+b)³
D)1/(ab(a+b))

Step-by-step solution

Let sin²θ = x and cos²θ = y. Then x+y=1. The given equation becomes x²/a + y²/b = 1/(a+b).
Substitute y = 1-x: x²/a + (1-x)²/b = 1/(a+b). This simplifies to (a+b)²x² - 2a(a+b)x + a² = 0, which is ((a+b)x - a)² = 0.
From this, (a+b)x = a, so x = a/(a+b). Thus, sin²θ = a/(a+b) and cos²θ = 1 - a/(a+b) = b/(a+b).
Now, substitute these values into the expression to be found: (sin²θ)⁴/a³ + (cos²θ)⁴/b³ = (a/(a+b))⁴/a³ + (b/(a+b))⁴/b³ = a/(a+b)⁴ + b/(a+b)⁴ = (a+b)/(a+b)⁴ = 1/(a+b)³.

Answer: 1/(a+b)³

Practice questions on Trigonometry

Q1.easy
If tan (A + B) = 1 and sin (A - B) = 1/2, where A and B are acute angles, what is the value of A?
1. A)30°
2. B)37.5°
3. C)45°
4. D)60°
Show answer
Answer: 37.5°
Hint: Use the inverse trigonometric functions to find the values of (A+B) and (A-B) first, then solve the resulting system of linear equations.
Q2.easy
If sin θ + cos θ = p and sin θ - cos θ = q, what is the value of p² + q²?
1. A)1
2. B)2 sin θ cos θ
3. C)2
4. D)sin²θ - cos²θ
Show answer
Answer: 2
Hint: Square both given equations and then add them. Remember the fundamental identity sin²θ + cos²θ = 1.
Q3.easy
A vertical pole stands on the ground. From a point on the ground 20 m away from the base of the pole, the angle of elevation of the top of the pole is 30°. A flag is hoisted at the top of the pole, and the angle of elevation of the top of the flag from the same point is 45°. What is the length of the flagstaff?
1. A)20(√3 - 1) m
2. B)20/√3 m
3. C)20(3 - √3)/3 m
4. D)20(√3 + 1)/3 m
Show answer
Answer: 20(3 - √3)/3 m
Hint: First, calculate the height of the pole using the angle of elevation to its top. Then, use the angle of elevation to the top of the flag to find the total height, and subtract to find the flagstaff's length.
Q4.medium
If x = a secθ + b tanθ and y = a tanθ + b secθ, where a ≠ b, find the value of x² - y².
1. A)a² + b²
2. B)a² - b²
3. C)2ab
4. D)a²b²
Show answer
Answer: a² - b²
Hint: Expand x² and y² and look for common terms or recognizable trigonometric identities to simplify the expression.
Q5.medium
If (sin A + cosec A)² + (cos A + sec A)² = k + tan²A + cot²A, then the value of k is:
1. A)7
2. B)5
3. C)6
4. D)8
Show answer
Answer: 7
Hint: Expand the squared terms and use the basic trigonometric identities involving sin, cos, cosec, and sec to simplify the expression.
Q6.medium
What is the value of tan 1° tan 2° tan 3° ... tan 89°?
1. A)0
2. B)√3
3. C)1
4. D)Cannot be determined
Show answer
Answer: 1
Hint: Consider the property of trigonometric ratios of complementary angles. Look for pairs of terms that can simplify.
Q7.hard
A vertical pole stands on a horizontal plane. The angle of elevation of the top of the pole from a point A on the plane is θ. A man walks 'd' meters from A directly towards the pole to point B. From B, the angle of elevation of the top of the pole is 2θ. He then walks 'd' meters further from B towards the pole to point C. From C, the angle of elevation of the top of the pole is 3θ. Find the height of the pole.
1. A)d√3
2. B)d/2
3. C)2d
4. D)d
Show answer
Answer: d
Hint: Set up equations using the cotangent of the angles of elevation and the given distances. Look for a relationship between cot θ, cot 2θ, and cot 3θ.
Q8.hard
If (sec θ + tan θ)(sec φ + tan φ)(sec ψ + tan ψ) = (sec θ - tan θ)(sec φ - tan φ)(sec ψ - tan ψ), then what is the value of (sec θ + tan θ)(sec φ + tan φ)(sec ψ + tan ψ)?
1. A)0
2. B)1
3. C)-1
4. D)2
Show answer
Answer: 1
Hint: Recall the fundamental trigonometric identity relating secant and tangent functions. How does (sec x - tan x) relate to (sec x + tan x)?
Q9.hard
What is the maximum value of sin θ + cos θ + sin 2θ?
1. A)1 + √2
2. B)1
3. C)2
4. D)(3+√2)/2
Show answer
Answer: 1 + √2
Hint: Consider the substitution t = sin θ + cos θ. How does sin 2θ relate to t?

These are 9 of the 60 questions available for Trigonometry. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.