By William J. Bruce, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)

ISBN-10: 0080103111

ISBN-13: 9780080103112

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**Extra resources for Analytic Trigonometry**

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The exsec2 Θ pression , 2 ±Vl - x can be transformed by letting x = sin Θ to get , which gives or sec Θ. Thus it is clear cos0 ±Vl - sin2 Θ that, by a suitable choice of the trigonometric function, many such transformations are possible and in some cases the form is simpler. Some care is necessary in transforming functions such as —. If we let x = sin 0, we get — = = = = = = # Vl - x2 Vl - sin2 0 Vcos2 0 is positive, we get when cos 0 > 0 and Since — cos 0 Vl — x2 when cos 0 < 0 or sec 0 when sec 0 > 0 but — sec 0 cos 0 1 when sec 0 < 0.

For example, since sin (90° - Θ) = cos 0, we have cos 60° = sin 30°, cos 70° = sin 20°, cos 45° = sin 45°, and so on. The same is true for the tangent and cotangent. 1 1. Find without using tables the value of sin 75°. 2. ), where a and ß are acute. 3. Show that Λ/Ϊ sin (0 - 45°) = sin 0 - cos 0 4. Reduce the following: (a) sin (270° + 0). (b) cos (180° - 0). (c) tan (270° - 0). 5. Express as cofunctions of the corresponding complementary angle: (a) sin 85°. (b) cos 72°. (c) tan 55°. 6. Express as functions of 2 a and 3 β: (a) sin (2 a + 3 β).

Express cos 4 Θ in terms of sin 2 0. cos 4 0 = 1 - 2 sin2 2 0. Example 2. Express sin2 2 0 in terms of cos 4 0. sin2 2 (9 = - (1 - cos 4 0). 56 ANALYTIC TRIGONOMETRY Example 3. 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.

### Analytic Trigonometry by William J. Bruce, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)

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