Grade 10 Trigonometry Practice: Missing Sides and Angles Explained

For a right-triangle problem, choose sine, cosine or tangent by identifying the sides relative to the angle: sine uses opposite and hypotenuse, cosine uses adjacent and hypotenuse, and tangent uses opposite and adjacent. Use an inverse trigonometric function when the unknown is the angle.

This original Grade 10 practice set includes eight worked questions. Unless stated otherwise, each triangle is right-angled, angles are in degrees, and lengths use the same unit. It is a focused practice resource, not a complete course test.

How do I choose sin, cos or tan?

Label the hypotenuse first: it is opposite the right angle. Choose the acute reference angle, then label its opposite side and its adjacent side. The adjacent side here means the non-hypotenuse side touching the reference angle.

Sine

• Relationship: sin θ = opposite / hypotenuse
• Use when these sides are involved: Opposite and hypotenuse

Cosine

• Relationship: cos θ = adjacent / hypotenuse
• Use when these sides are involved: Adjacent and hypotenuse

Tangent

• Relationship: tan θ = opposite / adjacent
• Use when these sides are involved: Opposite and adjacent

The labels opposite and adjacent change if you switch reference angles. The hypotenuse does not. Choose the ratio from the labelled sides, not from the appearance or orientation of the drawing.

Eight practice questions

• 1. A right triangle has hypotenuse 10 cm and an acute angle of 30°. Find the side opposite that angle.
• 2. An acute angle is 60° and its adjacent side is 7 cm. Find the hypotenuse.
• 3. The sides opposite and adjacent to an acute angle are 6 cm and 8 cm. Find the angle.
• 4. A ladder of length 5.00 m makes a 70.0° angle with level ground. How high up a vertical wall does it reach? Assume the ladder is straight and the wall is perpendicular to the ground.
• 5. A ramp rises 0.90 m over a horizontal distance of 6.00 m. Find its angle to the horizontal.
• 6. A right triangle has hypotenuse 13 cm and one leg 5 cm. Find the other leg, then the acute angle opposite the 5 cm leg.
• 7. A student writes sin⁻¹(0.5) = 1/sin(0.5). Explain the error and find sin⁻¹(0.5) in degrees.
• 8. In a non-right triangle, two sides are 5 cm and 7 cm with an included angle of 60°. Find the third side. Explain why SOHCAHTOA cannot be applied directly to the whole triangle.

Worked solutions

1. Find an opposite side

sin 30° = x/10, so x = 10 sin 30° = 5 cm. The opposite leg is shorter than the hypotenuse, which is a useful reasonableness check.

2. Find a hypotenuse

cos 60° = 7/h. Multiply both sides by h and divide by cos 60°: h = 7/cos 60° = 14 cm. Multiplying 7 by cos 60° would make the hypotenuse shorter than a leg and reveal a setup error.

3. Find an angle from two legs

tan θ = 6/8 = 0.75. Therefore θ = arctan(0.75) ≈ 36.9°. Use degree mode. This is the angle opposite the 6 cm leg; the other acute angle is different.

4. Model the ladder situation

The ladder is the hypotenuse and the vertical height is opposite the ground angle. h = 5.00 sin 70.0° ≈ 4.70 m. The result is a mathematical model of the stated geometry, not advice on safe ladder placement.

5. Use rise and horizontal run

tan θ = 0.90/6.00 = 0.15. Thus θ = arctan(0.15) ≈ 8.53°. The horizontal distance is the adjacent leg, not the length of the ramp surface.

6. Combine Pythagoras with trigonometry

The other leg is √(13² − 5²) = √144 = 12 cm. For the angle opposite the 5 cm leg, sin θ = 5/13, so θ ≈ 22.6°. The equation 5² + 12² = 13² confirms the side lengths.

7. Distinguish inverse from reciprocal

In calculator notation, sin⁻¹ means arcsine: the inverse function used to find an angle from a sine value. It does not mean 1/sin. Since sin 30° = 0.5, sin⁻¹(0.5) = 30° for the principal value.

8. Recognize a different triangle model

Use the cosine law: c² = 5² + 7² − 2(5)(7)cos 60° = 25 + 49 − 35 = 39. Therefore c = √39 cm ≈ 6.24 cm. SOHCAHTOA describes right-triangle ratios; the given whole triangle has no stated right angle. A suitable construction could create right triangles, but the cosine law handles the given information directly.

Why do word problems feel harder than the examples?

The extra task is choosing a model. Before calculating, draw the situation, mark the angle from the correct reference line, and distinguish vertical height, horizontal distance and sloping length. Then label only the known and requested sides.

If you can follow a worked solution but cannot start a fresh question, practise the setup separately: choose the ratio and write the equation without calculating. That reveals whether the difficulty is interpretation, ratio selection or arithmetic.

Continue your Grade 10 practice

Read the quadratic equations guide, explore Grade 10 math tutoring, or book a free one-hour assessment.

Curriculum reference: Ontario Grades 9–10 Mathematics. Questions and explanations are original.