GRADE 10 · A4 PRINT VIEW
Introduction to Trigonometry · Answer guide
Use your browser Print command and select A4. Questions and answers have separate pages.
Lesson 1 · Build ratios from a right triangle
Choose sine, cosine and tangent relative to an acute angle.
For acute angle θ, opposite = 3, adjacent = 4, hypotenuse = 5. Find sin θ.
Answer: 3/5. sin uses opposite/hypotenuse, giving 3/5.
For acute angle θ, opposite = 3, adjacent = 4, hypotenuse = 5. Find cos θ.
Answer: 4/5. cos uses adjacent/hypotenuse, giving 4/5.
For acute angle θ, opposite = 5, adjacent = 12, hypotenuse = 13. Find tan θ.
Answer: 5/12. tan uses opposite/adjacent, giving 5/12.
Discuss: Draw two similar right triangles, one twice as large. Show that their sine ratios for matching angles are equal.
Teaching note: Opposite and adjacent depend on the chosen acute angle. The hypotenuse does not. Open responses need discussion; they are not automatically marked.
Lesson 2 · Derive familiar angle values
Connect 30°, 45° and 60° values to special triangles.
What is sin 30°?
Answer: 1/2. The standard right-triangle ratios give sin 30° = 1/2. For 30°/60° use a 1:√3:2 triangle; for 45° use 1:1:√2.
What is cos 60°?
Answer: 1/2. The standard right-triangle ratios give cos 60° = 1/2. For 30°/60° use a 1:√3:2 triangle; for 45° use 1:1:√2.
What is tan 45°?
Answer: 1. The standard right-triangle ratios give tan 45° = 1. For 30°/60° use a 1:√3:2 triangle; for 45° use 1:1:√2.
Discuss: Make a table of sin, cos and tan at 0°, 30°, 45°, 60° and 90°. Label the undefined entry explicitly.
Teaching note: tan 90° is undefined, not zero. Exact surds avoid rounding errors in later steps. Open responses need discussion; they are not automatically marked.
Lesson 3 · Use identities with conditions
Derive sin²θ + cos²θ = 1 and use it to find a missing ratio.
θ is acute and sin θ = 3/5. What is cos θ?
Answer: 4/5. cos²θ = 1 − sin²θ = 1 − (3/5)² = 16/25. Since θ is acute, cos θ is positive, giving 4/5.
θ is acute and sin θ = 5/13. What is cos θ?
Answer: 12/13. cos²θ = 1 − sin²θ = 1 − (5/13)² = 144/169. Since θ is acute, cos θ is positive, giving 12/13.
θ is acute and sin θ = 8/17. What is cos θ?
Answer: 15/17. cos²θ = 1 − sin²θ = 1 − (8/17)² = 225/289. Since θ is acute, cos θ is positive, giving 15/17.
Discuss: Explain why 1 − sin²θ equals cos²θ, while 1 − sin θ does not generally equal cos θ.
Teaching note: sin²θ means (sin θ)²; it is not sin(θ²). Taking a square root may require a sign condition. Open responses need discussion; they are not automatically marked.
Final check · Set 1
Try one set after the lessons. Use the other set after reviewing any errors.
For acute angle θ, opposite = 7, adjacent = 24, hypotenuse = 25. Find cos θ.
24/25. cos uses adjacent/hypotenuse, giving 24/25.
For acute angle θ, opposite = 20, adjacent = 21, hypotenuse = 29. Find tan θ.
20/21. tan uses opposite/adjacent, giving 20/21.
What is cos 30°?
√3/2. The standard right-triangle ratios give cos 30° = √3/2. For 30°/60° use a 1:√3:2 triangle; for 45° use 1:1:√2.
What is sin 45°?
1/√2. The standard right-triangle ratios give sin 45° = 1/√2. For 30°/60° use a 1:√3:2 triangle; for 45° use 1:1:√2.
θ is acute and sin θ = 4/5. What is cos θ?
3/5. cos²θ = 1 − sin²θ = 1 − (4/5)² = 9/25. Since θ is acute, cos θ is positive, giving 3/5.
θ is acute and sin θ = 12/13. What is cos θ?
5/13. cos²θ = 1 − sin²θ = 1 − (12/13)² = 25/169. Since θ is acute, cos θ is positive, giving 5/13.
Final check · Set 2
Try one set after the lessons. Use the other set after reviewing any errors.
For acute angle θ, opposite = 9, adjacent = 12, hypotenuse = 15. Find sin θ.
3/5. sin uses opposite/hypotenuse, giving 3/5.
For acute angle θ, opposite = 12, adjacent = 16, hypotenuse = 20. Find cos θ.
4/5. cos uses adjacent/hypotenuse, giving 4/5.
What is tan 30°?
1/√3. The standard right-triangle ratios give tan 30° = 1/√3. For 30°/60° use a 1:√3:2 triangle; for 45° use 1:1:√2.
What is tan 60°?
√3. The standard right-triangle ratios give tan 60° = √3. For 30°/60° use a 1:√3:2 triangle; for 45° use 1:1:√2.
θ is acute and sin θ = 15/17. What is cos θ?
8/17. cos²θ = 1 − sin²θ = 1 − (15/17)² = 64/289. Since θ is acute, cos θ is positive, giving 8/17.
θ is acute and sin θ = 24/25. What is cos θ?
7/25. cos²θ = 1 − sin²θ = 1 − (24/25)² = 49/625. Since θ is acute, cos θ is positive, giving 7/25.
Original learning companion, not full textbook exercise solutions. Covers right-triangle ratios, standard acute-angle values and basic identities. It does not exhaust all identity proofs or textbook exercises.