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Trigonometry
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8.1 Trigonometric Ratios
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Trigonometry means measurement of triangles. Consider its parts: tri means three, gon means angle, and metry means measure. Thus, in trigonometry we study the measurement of triangles. The following ratios relate the sides and acute angles of a right triangle: 1. Tangent ratio: The tangent (abbreviated tan ) of an acute angle equals the length of the leg opposite the angle divided by the length of the leg adjacent to the angle. 2. Sine ratio: The sine (abbreviated sin ) of an acute angle equals the length of the leg opposite the angle divided by the length of the hypotenuse. 3. Cosine ratio: The cosine (abbreviated cos ) of an acute angle equals the length of the leg adjacent to the angle divided by the length of the hypotenuse. Thus in right triangle ABC of Fig. 8-1,
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Fig. 8-1
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tan A sin A
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length of leg opposite A length of leg adjacent to A length of leg opposite A length of hypotenuse length of leg adjacent to A length of hypotenuse a c
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length of leg opposite B length of leg adjacent to B length of leg opposite B length of hypotenuse length of leg adjacent to B length of hypotenuse b c
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If A and B are the acute angles of a right triangle, then sin A cos B cos A sin B tan A 1 tan B tan B 1 tan A
A scientific calculator can compute the sine, cosine, and tangent of an angle with the SIN, COS, and TAN buttons, respectively. Make sure the calculator is set to degrees (DEG). For those without a calculator, a table of sines, cosines, and tangents is in the back of this book.
CHAPTER 8 Trigonometry
SOLVED PROBLEMS
Using the table of sines, cosines, and tangents The following values were taken from a table of sines, cosines, and tangents. State, in equation form, what the values on the first three lines mean. Then use the table at the back of this book to complete the last line.
(a) (b) (c) (d)
Angle 1 30 60
Sine 0.0175 0.5000 0.8660
Cosine 0.9998 0.8660 0.5000 0.3420
Tangent 0.0175 0.5774 1.7321
Solutions
(a) sin 1 (b) sin 30 (c) sin 60 0.0175; cos 1 0.5000; cos 30 0.8660; cos 60 0.9998; tan 1 0.8660; tan 30 0.5000; tan 60 0.0175 0.5774 1.7321
(d) In the table of trigonometric functions, the cosine value 0.3420 is on the 70 line; hence, the angle measures 70 . Then, from the table, sin 70 0.9397 and tan 70 2.7475.
Finding angle measures to the nearest degree Find the measure of x to the nearest degree if (a) sin x 0.9235; (b) cos x (c) tan x 25 / 10 or 0.2236. Use the table of trigonometric functions.
Solutions
Differences (a) sin 68 0.9272 sin x 0.9235 sin 67 0.9205 (b) cos 32 0.8480 cos x 0.8400 cos 33 0.8387 (c) tan 13 0.2309 tan x 0.2236 tan 12 0.2126 S 0.0037 S 0.0030 S 0.0080 S 0.0013 S 0.0073 S 0.0110 Since sin x is nearer to sin 67 , m x
21 25
or 0.8400;
67 to the nearest degree.
Since cos x is nearer to cos 33 , m x
33 to the nearest degree.
Since tan x is nearer to tan 13 , m x
13 to the nearest degree.
With a calculator, the above answers can be found with the inverse sine (sin 1), inverse cosine (cos 1), and inverse tangent (tan 1). These usually require first pressing the 2nd or INV button and then SIN, COS, or TAN.
Finding trigonometric ratios For each right triangle in Fig. 8-2, find the trigonometric ratios of each acute angle.
CHAPTER 8 Trigonometry
Fig. 8-2
Solutions
Formulas
tan A tan B sin A sin B cos A cos B a b b a a c b c b c a c
(a) a
3, b
tan A tan B sin A sin B cos A cos B
4, c
3 4 4 3 3 5 4 5 4 5 3 5
(b) a
6, b
6 8 8 6 6 10 8 10
8, c
3 4 4 3 3 5 4 5 4 5 3 5
(c) a
5, b
tan A tan B sin A sin B cos A cos B
12, c
5 12 12 5 5 13 12 13 12 13 5 13
tan A tan B sin A sin B cos A cos B
8 10 6 10
Finding measures of angles by trigonometric ratios Find the measure of angle A, to the nearest degree, in each part of Fig. 8-3.
Fig. 8-3