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CHAPTER 3 Trigonometric Functions of an Acute Angle
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opposite side hypotenuse adjacent side hypotenuse opposite side adjacent side adjacent side opposite side hypotenuse adjacent side hypotenuse opposite side
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sin A cos A tan A
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3.2 Trigonometric Functions of Complementary Angles
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The acute angles A and B of the right triangle ABC are complementary; that is, A we have sin B cos B tan B b/c a/c b/a cos A sin A cot A cot B sec B csc B a/b c/a c/b tan A csc A sec A B 90 . From Fig. 3.1,
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These relations associate the functions in pairs sine and cosine, tangent and cotangent, secant and cosecant each function of a pair being called the cofunction of the other. Thus, any function of an acute angle is equal to the corresponding cofunction of the complementary angle.
EXAMPLE 3.1 Find the values of the trigonometric functions of the angles of the right triangle ABC in Fig. 3.3.
Fig. 3.3
sin A cos A tan A sin B cos B tan B
opposite side hypotenuse adjacent side hypotenuse opposite side adjacent side opposite side hypotenuse ajcacent side hypotenuse opposite side adjacent side
a c b c a b b c a c b a
60 109 91 109 60 91 91 109 60 109 91 60
csc A sec A cot A csc B sec B cot B
hypotenuse opposite side hypotenuse adjacent side adjacent side opposite side hypotenuse opposite side hypotenuse adjacent side adjacent side opposite side
c a c b b a c b c a a b
109 60 109 91 91 60 109 91 109 60 60 91
3.3 Trigonometric Functions of 30 , 45 , and 60
The special acute angles 30 , 45 , and 60 (see App. 1, Geometry) have trigonometric function values that can be computed exactly. The following results are obtained in Probs. 3.8 and 3.9. For each fraction that had an irrational number denominator, only the equivalent fraction with a rational number denominator is stated in the table.
CHAPTER 3 Trigonometric Functions of an Acute Angle
Angle 30 45 60
1 2 12 2 13 3
13 2 12 2 1 2
13 3
cot 23 1
13 3
2 13
csc 2 22
2 13
1 23
22 2
3.4 Trigonometric Function Values
For many application problems, values of trigonometric functions are needed for angles that are not special angles. These values may be found in tables of trigonometric functions or by using a scientific calculator. Problems 3.10 to 3.15 illustrate a number of simple applications of trigonometric functions. For these problems, a two-decimal-place table is included below.
Angle u 15 20 30 40 45 50 60 70 75 sin u 0.26 0.34 0.50 0.64 0.71 0.77 0.87 0.94 0.97 cos u 0.97 0.94 0.87 0.77 0.71 0.64 0.50 0.34 0.26 tan u 0.27 0.36 0.58 0.84 1.00 1.19 1.73 2.75 3.73 cot u 3.73 2.75 1.73 1.19 1.00 0.84 0.58 0.36 0.27 sec u 1.04 1.06 1.15 1.31 1.41 1.56 2.00 2.92 3.86 csc u 3.86 2.92 2.00 1.56 1.41 1.31 1.15 1.06 1.04
When using a calculator to find values for trigonometric functions, be sure to follow the procedure indicated in the instruction manual for your calculator. In general the procedure is (1) make sure the calculator is in degree mode, (2) enter the number of degrees in the angle, (3) press the key for the trigonometric function wanted, and (4) read the function value from the display.
EXAMPLE 3.2 Find tan 15 using a calculator. With the calculator in degree mode, enter 15 and press the (tan) key. The number 0.267949 will appear on the display; thus tan 15 0.267949. The number of digits that are displayed depends on the calculator used, but most scientific calculators show at least six digits. In this book if the value displayed on a calculator is not exact, it will be rounded to six digits when stated in a problem or example. Rounding procedures for final results will be introduced as needed.
Using a calculator to find an acute angle when the value of a trigonometric function is given requires the use of the inverse (inv) key or the second function (2nd) key. The value of the function is entered, the (inv) key is pressed, and then the trigonometric function key is pressed. The degree mode is used to get answers in degree measure.
EXAMPLE 3.3 Find acute angle A when sin A 0.2651. With the calculator in degree mode, enter .2651 and press the (inv) key and the (sin) key. The number 15.3729 on the display is the degree measure of acute angle A. Thus to the nearest degree A 15 .
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