Trigonometry Worksheet — GCSE — Intermediate

Trigonometry worksheet for Key Stage 4 / GCSE (Ages 14-16). Intermediate level maths practice, aligned to the UK National Curriculum. Print-ready with answer key included.

Maths•Key Stage 4 / GCSE (Ages 14-16)•Trigonometryintermediate

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Exploring Trigonometry: Intermediate Challenges

Key Stage 4 / GCSE, Ages 14-16, Intermediate Level

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Instructions: This worksheet focuses on developing your understanding of trigonometry. You will solve problems involving right-angled triangles, calculate angles, and work with trigonometric ratios. Ensure you show all your workings and use a calculator where necessary.

Trigonometric Ratios

Calculate the trigonometric ratios for the given angles in right-angled triangles. Use a calculator to find the sine, cosine, and tangent values to two decimal places.

1. Find sin(30°), cos(30°), and tan(30°).

2. Find sin(45°), cos(45°), and tan(45°).

3. Find sin(60°), cos(60°), and tan(60°).

Solving Right-Angled Triangles

Use trigonometric ratios to find the unknown sides in each right-angled triangle. Assume all angles are in degrees and round your answers to two decimal places.

1. In triangle ABC, angle A = 30°, side a = 5 cm. Find side b.

2. In triangle DEF, angle D = 45°, side d = 7 cm. Find side e.

3. In triangle GHI, angle G = 60°, side g = 10 cm. Find side h.

Finding Angles Using Inverse Trigonometric Functions

Calculate the angles in degrees using inverse trigonometric functions. Round your answers to one decimal place.

1. If sin θ = 0.5, find θ.

2. If cos θ = 0.707, find θ.

3. If tan θ = 1.732, find θ.

Real-World Applications of Trigonometry

Solve these real-world problems using trigonometry. Clearly show your steps and calculations.

1. A ladder is leaning against a wall, forming a 75° angle with the ground. If the ladder is 10 m long, how high does it reach on the wall?

2. A kite is flying at a height of 50 m, and the angle of elevation from the observer is 30°. How far is the observer from the kite?

Trigonometric Identities

Verify the following trigonometric identities. Show all steps in your working.

1. Prove that sin²θ + cos²θ = 1 for θ = 45°.

2. Prove that 1 + tan²θ = sec²θ for θ = 30°.

Graphing Trigonometric Functions

Sketch the graphs of the following trigonometric functions over the interval 0° to 360°. Label key points on your graphs.

1. y = sin x

2. y = cos x

3. y = tan x

Solving Trigonometric Equations

Solve the following trigonometric equations for 0° ≤ x < 360°. Provide all solutions within the given range.

1. Solve sin x = 0.5

2. Solve cos x = 0.866

3. Solve tan x = 1

Extended Writing: Trigonometry in Architecture

Write an essay discussing the importance of trigonometry in architecture. Include examples of how architects use trigonometric principles in design and construction.

Discuss the role of trigonometry in architecture, providing examples of its application in building design and structural engineering. [10 marks]

Answer Key

Activity 1: sin(30°) = 0.5 ; cos(30°) = 0.87 ; tan(30°) = 0.58

Activity 2: 2.5 cm ; 4.95 cm ; 5.77 cm

Activity 3: 30° ; 45° ; 60°

Activity 4: 9.66 m ; 86.6 m

Activity 5: Verified ; Verified

Activity 6: Graphs sketched with key points labeled

Activity 7: x = 30°, 150° ; x = 30°, 330° ; x = 45°, 225°

Activity 8: Essay on trigonometry in architecture with examples