Trigonometric Functions

Trigonometry began as the study of the sides and angles of triangles. This chapter extends those ideas to angles of any size, positive or negative, so that sine, cosine and the other functions become functions of a real number. It begins with the measurement of angles in degrees and radians, using pi radians = 180 degrees and the relation l = r(theta) between arc length, radius and the angle at the centre. The trigonometric functions are then defined using the unit circle, and you will learn their signs in the four quadrants, their domains and ranges, and their periodic graphs. The chapter develops the identity sin^2 x + cos^2 x = 1, the values of the functions for angles such as pi/6, pi/4 and pi/3, and allied angles such as pi - x and pi + x. You will study the compound-angle formulas for sin(x + y), cos(x + y) and tan(x + y), and the multiple-angle formulas for sin 2x, cos 2x, tan 2x, sin 3x and cos 3x.

Detailed Worksheet: Trigonometric Functions Section A - Definitions (10 marks) 1. Convert 40 degrees 20 minutes into radian measure. (2 marks) 2. Find the value of sin 765 degrees. (2 marks) 3. Find the value of tan(19pi/3). (2 marks) 4. State the sign of sin, cos and tan in each of the four quadrants. (2 marks) 5. Write the domain and range of the sine and tangent functions. (2 marks) Section B - Calculations and Applications (15 marks) 6. If cos x = -1/2 and x lies in the third quadrant, find the values of the other five trigonometric functions. (3 marks) 7. Find the length of an arc of a circle of radius 21 cm that subtends an angle of 60 degrees at the centre. Use pi = 22/7. (3 marks) 8. Find the value of sin 75 degrees using a compound-angle formula. (3 marks) 9. Prove that 2sin^2(pi/6) + cosec^2(7pi/6) cos^2(pi/3) = 3/2. (3 marks) 10. A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second? (3 marks) Section C - Diagrams (10 marks) 11. Draw the unit circle and show how sin x and cos x are defined for an angle x in the second quadrant. (4 marks) 12. Sketch the graph of y = sin x for 0 <= x <= 2pi and mark its maximum and minimum values. (3 marks) 13. Sketch the graph of y = cos x for 0 <= x <= 2pi and state its period. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Prove that cos(x + y) = cos x cos y - sin x sin y and use it to find cos 15 degrees. (5 marks) 15. If sin x = 3/5 and x lies in the second quadrant, find sin 2x, cos 2x and tan 2x. (5 marks) 16. If tan x = -4/3 and x lies in the second quadrant, find sin(x/2), cos(x/2) and tan(x/2). (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Give exact values with surds where possible.