Trigonometry Worksheet — GCSE — Advanced

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

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

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Advanced Trigonometry Challenges

Key Stage 4 / GCSE (Ages 14-16) - Advanced Level

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Instructions: This worksheet is designed to challenge your understanding of advanced trigonometric concepts. You will solve problems involving angles, identities, and real-world applications. Ensure you show all your workings clearly.

Solving Trigonometric Equations

Solve each of the following trigonometric equations for x in the interval 0° ≤ x < 360°. Show all your workings and provide the general solution where applicable.

1. sin(x) = 0.5

2. cos(x) = -0.5

3. tan(x) = 1

4. 2sin(x) - 1 = 0

5. cos(2x) = 0

Trigonometric Identities

Prove the following trigonometric identities. Show each step of your working clearly.

1. Prove that sin²(x) + cos²(x) = 1

2. Prove that 1 + tan²(x) = sec²(x)

3. Prove that sin(2x) = 2sin(x)cos(x)

Real-World Applications

Apply your knowledge of trigonometry to solve these real-world problems. Use diagrams where necessary and show all your workings.

1. A ladder is leaning against a wall, making an angle of 60° with the ground. If the ladder is 10m long, how far is the base of the ladder from the wall?

2. A kite is flying at a height of 50m with the string making an angle of 30° with the ground. How long is the string?

Graphical Interpretation

Sketch the graphs of the following trigonometric functions for 0° ≤ x ≤ 360°. Label key points such as intercepts and maximum/minimum values.

1. y = sin(x)

2. y = cos(x)

3. y = tan(x)

Angle of Elevation and Depression

Solve the problems involving angles of elevation and depression. Draw diagrams to assist your solution and show all your workings.

1. From a point 30m away from the base of a tree, the angle of elevation to the top of the tree is 45°. Find the height of the tree.

2. A person standing on a cliff observes a boat at an angle of depression of 20°. If the cliff is 50m high, how far is the boat from the base of the cliff?