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Chapter 4: Problem 61

If you are given the lengths of the sides of a right triangle, describe how to find the sine of either acute angle.

Short Answer

Expert verified
The sine of an acute angle in a right triangle is calculated by dividing the length of the side opposite the angle by the length of the hypotenuse. This is represented mathematically as \(\sin(A) = \frac{a}{c}\)

Step by step solution

01

Understand Triangle's Nomenclature

Before proceeding, one must identify the parts of the right triangle. The longest side of the triangle, located opposite the right angle, is called the hypotenuse. The remaining two sides are known as the opposite and adjacent, relative to the angle of interest. If the triangle's sides are named 'a', 'b', and 'c', then by convention 'c' often represents the hypotenuse.
02

Sin Calculation

In a right triangle, the sine of an angle is calculated as the ratio of the opposite side to the hypotenuse. Suppose 'A' represents one of the acute angles of the right triangle, and 'a' represents the side opposite to it (and 'c' represents the hypotenuse), the sine of the angle 'A' is calculated using the formula: \(\sin(A) = \frac{a}{c}\)
03

Execution of the Calculation

Plug the lengths of sides 'a' and 'c' into the formula to get the sine of the angle.

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Most popular questions from this chapter

Define the sine of \(t\).

Use words (not an equation) to describe one of the quotient identities.

The height of the water, \(H,\) in feet, at a boat dock \(t\) hours after 6 A.M. is given by $$H=10+4 \sin \frac{\pi}{6} t$$ a. Find the height of the water at the dock at 6 A.M., 9 A.M., noon, 6 P.M., midnight, and 3 A.M. b. When is low tide and when is high tide? c. What is the period of this function and what does this mean about the tides?

Will help you prepare for the material covered in the first section of the next chapter. The exercises use identities, introduced in Section \(4.2,\) that enable you to rewrite trigonometric expressions so that they contain only sines and cosines: $$\begin{array}{ll} \csc x=\frac{1}{\sin x} & \sec x=\frac{1}{\cos x} \\ \tan x=\frac{\sin x}{\cos x} & \cot x=\frac{\cos x}{\sin x} \end{array}$$ Rewrite each expression by changing to sines and cosines. Then simplify the resulting expression. $$\tan x \csc x \cos x$$

Use the keys on your calculator or graphing utility for converting an angle in degrees, minutes, and seconds \(\left(D^{\circ} M^{\prime} S^{\prime \prime}\right)\) into decimal form, and vice versa. Convert each angle to \(D^{\circ} M^{\prime} S^{\prime \prime}\) form. Round your answer to the nearest second. $$50.42^{\circ}$$
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