Introduction to Trigonometry

Trigonometry studies the relationship between the sides and angles of a right triangle. In this chapter you will define the six trigonometric ratios of an acute angle A in a right triangle ABC, right-angled at B: sin A = BC/AC, cos A = AB/AC, tan A = BC/AB, and their reciprocals cosec A, sec A and cot A. You will see that these ratios depend only on the angle, not on the size of the triangle, because triangles with the same angles are similar. You will find the exact values of the ratios for 0, 30, 45, 60 and 90 degrees using an equilateral triangle and an isosceles right triangle, and learn which ratios are undefined at 0 or 90 degrees. Given one ratio, you will find all the others with Pythagoras theorem. Finally you will prove the three fundamental identities, sin^2 A + cos^2 A = 1, 1 + tan^2 A = sec^2 A and 1 + cot^2 A = cosec^2 A, and use them to simplify expressions and prove other identities.

Detailed Worksheet: Introduction to Trigonometry Section A - Definitions (10 marks) 1. In a right triangle ABC, right-angled at B, define sin A, cos A and tan A in terms of the sides of the triangle. (2 marks) 2. Define cosec A, sec A and cot A, and write how each is related to sin A, cos A and tan A. (2 marks) 3. Explain why the trigonometric ratios of an angle do not depend on the lengths of the sides of the right triangle. (2 marks) 4. Write the values of sin, cos and tan for 30, 45 and 60 degrees. (2 marks) 5. State the three fundamental trigonometric identities and the range of A for which each is valid. (2 marks) Section B - Calculations and Applications (15 marks) 6. Given 15 cot A = 8, find sin A and sec A. (3 marks) 7. In triangle PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P. (3 marks) 8. Evaluate: (i) sin 60 cos 30 + sin 30 cos 60 (ii) 2 tan^2 45 + cos^2 30 - sin^2 60 (iii) (5 cos^2 60 + 4 sec^2 30 - tan^2 45)/(sin^2 30 + cos^2 30), all angles in degrees. (3 marks) 9. If tan (A + B) = sqrt(3) and tan (A - B) = 1/sqrt(3), with 0 < A + B <= 90 degrees and A > B, find A and B. (3 marks) 10. In triangle ABC, right-angled at B, AB = 5 cm and angle ACB = 30 degrees. Determine the lengths of the sides BC and AC. (3 marks) Section C - Diagrams (10 marks) 11. Draw a right triangle ABC, right-angled at B, and mark the side opposite to angle A, the side adjacent to angle A and the hypotenuse. Write sin A, cos A and tan A and also sin C, cos C and tan C from your figure. (4 marks) 12. Draw an equilateral triangle ABC of side 2a with the perpendicular AD from A to BC. Use the figure to obtain the values of sin 30, cos 30, sin 60 and cos 60 degrees. (3 marks) 13. Draw a right triangle ABC, right-angled at B, with angle A = 45 degrees. Taking BC = a, use the figure to find sin 45, cos 45 and tan 45 degrees. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Prove the identity (sin A + cosec A)^2 + (cos A + sec A)^2 = 7 + tan^2 A + cot^2 A, stating which fundamental identities you use. (5 marks) 15. State whether the following are true or false, and justify your answer: (i) the value of tan A is always less than 1 (ii) sec A = 12/5 for some value of angle A (iii) cos A is the abbreviation used for the cosecant of angle A (iv) cot A is the product of cot and A (v) sin theta = 4/3 for some angle theta. (5 marks) 16. Express the ratios cos A, tan A and sec A in terms of sin A. Use your results to find cos A, tan A and sec A when sin A = 3/5, and verify that sec^2 A - tan^2 A = 1. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. All angles are in degrees. Draw a figure for every question involving a triangle and show each step of simplification.