By William J. Bruce, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)
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Similarly, cos Θ = - approaches - = 0 y i and cos 90° = 0, tan Θ = - approaches - which is undefined and hence tan 90° is undefined, and so on. This is a dynamic approach whereas the first method employed is a static approach. With a little practice, the latter method is found to be faster than the former. Example 2. Find the functions of 180° (π). We use method 2 (Fig. 14) noting that y tends to zero whereas x tends to — 1 when r = 1 and Θ tends to 180°. Then sin 180° = 0, cos 180° = - 1, tan 180° = 0, cot 180° = (undefined), sec 180° = - 1, csc 180° = (undefined).
Sin 0 = 1 3. sin 0 = - \ 4. sin 0 = — 2 5. cos Θ = £ y/ï 6. cos 0 = - \ Λ/3 Λ/2 TRIGONOMETRIC FUNCTIONS OF ANGLES 7. cos 0 = 0 10. tan 0 = — \ 8. cos 0 = - 1 45 9. tan 0 = 1 \/f Find the values of 0, such that 0° < 0 < 360°, which satisfy each of the equations: 11. sin2 0 = i 12. cos2 0 = i 13. 3 tan2 0 = 1 14. 3 sec2 0 = 4 15. cot2 0 = 1 16. tan2 0 = 2 sin2 0 17. 2 tan2 0 = sec2 0 18. sin 0 = cos 0 Solve the following equations for 0 in radian measure, such that 0 < 0 < 2 π: 19Γ2 sin2 0 - cos^0 = 1 20.
Express cos4 Θ in first powers. cos4 0 = cos2 θ cos2 Θ = - (1 + cos 2 Θ) - (1 + cos 2 Θ) = - (1 + 2 cos 2 0 + cos2 2 0) = ~ | l + 2 cos 2 0 + ί (1 + cos 4 0)1 - I f 3 + 2 cos 2 0 + 1- cos 4 N0 . 2 1. Express in functions of half the angle: (a) sin 4 Θ. (b) sin k Θ (c) cos 6 0, in sines. (d) cos 3 0, in cosines. (e) sin 3 Θ. (f) cos k Θ, in sines. (g) tan 4 Θ. (h) tan £ 0. 2. Express in functions of twice the angle: (a) sin2 4 Θ. (d) cos 2 0. (b) sin 2 0. (e) sin2fc0. 2 (c) cos 4 0. (f ) cos2 k 0.