By William J. Bruce, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)
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Extra resources for Analytic Trigonometry
Hence 90° - y? and y? are complementary angles. Since sin (90° — ß) is equal to the cosine FUNCTIONS OF SUMS A N D RELATED FUNCTIONS 51 of the complementary angle ß, we call the cosine the complementary function of the sine or in short the cofunction. Thus the sine and cosine are called cofunctions. Similarly, the tangent and cotangent are cofunctions. In addition to the above sets, we include 4. Tangents and cotangents of sums and differences tan (a + ß) tan a + tan ß 1 - tan oc tan ß tan (a - ß) tan ce — tan ß 1 + tan a tan ß cot (a + ß) cot a cot ß — 1 cot ce + cot ß cot (a - ß) = cot a cot ß + 1 cot ß — cot α We prove the first of these only.
If cot Θ = 3, find all other functions of Θ. 5. From a diagram determine the other four functions of 0 if tan Θ = f, sec 0 = — 4. 6. If sin 0 = 4 and tan Θ is positive, find all the functions of 0. 7. If sin 0 = — ~2 and cos Θ is positive, find all the functions of Θ. 8. If sin Θ = f and r = 36, find x. 9. If cos Θ = f and x = 10,find>\ 10. If csc Θ = b and tan 0 = c, find sin 0 and cot 0. 9. Tables of Trigonometric Functions We have seen that functions of certain special angles can be determined easily by constructing and labelling appropriately the reference triangle.
T 11. csc 0 - cot 0 -tan ( 4 ) sin 2 0 12. — — = tan 0. 1 + cos 2 0 13. sec 2 0 1 - 2 sin2 0 * 14. (sin 0 - cos 0)2 = 1 - sin 2 0. 15. tan 2 0 cot 0 16. 1 - sin 2 0 cos 2 0 2 1 - tan2 0 1 - tan 0 1 + tan 0 ' Solve the following equations for 0 < 0 < 2 π: 17. sin 2 0 = i λ/3" 18. cos 2 0 = - -^- . 19. 1 - 2 sin2 2 0 = h (Show two methods). 20. 2 cos2 0 — 1 = - 2. (Show two methods). 21. sin (2 0 + Ή 22. tan (l 0 - ^ j = 1. Λ= - i 23. (sin 2 0 - 1) (sin 0 + 1) = 0. 24. 2 sin2 ( | ) - 3 sin (JA +1=0.
Analytic Trigonometry by William J. Bruce, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)