1.4

Trigonometry

Scientists use mathematics to help them describe how the physical universe works, and trigonometry is an important branch of mathematics. Three trigonometric functions are utilized throughout this text. They are the sine, the cosine, and the tangent of the angle (Greek theta), abbreviated as , and , respectively. These functions are defined below in terms of the symbols given along with the right triangle in Figure 1-5.

Figure 1-5 A right triangle.

| Definition of sin θ, cos θ, and tan θ

| (1.1)

| (1.2)

| (1.3)

h = length of the hypotenuse of a right triangle
ho = length of the side opposite the angle
ha = length of the side adjacent to the angle

The sine, cosine, and tangent of an angle are numbers without units, because each is the ratio of the lengths of two sides of a right triangle. Example 4 illustrates a typical application of Equation 1.3.

| Example 4 | | | Using Trigonometric Functions

On a sunny day, a tall building casts a shadow that is 67.2 m long. The angle between the sun’s rays and the ground is , as Figure 1-6 shows. Determine the height of the building.

Figure 1-6 From a value for the angle and the length ha of the shadow, the height ho of the building can be found using trigonometry.

Reasoning We want to find the height of the building. Therefore, we begin with the colored right triangle in Figure 1-6 and identify the height as the length ho of the side opposite the angle . The length of the shadow is the length ha of the side that is adjacent to the angle . The ratio of the length of the opposite side to the length of the adjacent side is the tangent of the angle , which can be used to find the height of the building.
Solution We use the tangent function in the following way, with and :

| (1.3)

The value of tan 50.0° is found by using a calculator.

The sine, cosine, or tangent may be used in calculations such as that in Example 4, depending on which side of the triangle has a known value and which side is asked for. However, the choice of which side of the triangle to label ho (opposite) and which to label ha (adjacent) can be made only after the angle is identified.

Problem solving insight

Often the values for two sides of the right triangle in Figure 1-5 are available, and the value of the angle is unknown. The concept of inverse trigonometric functions plays an important role in such situations. Equations 1.4, 1.5 and 1.6 give the inverse sine, inverse cosine, and inverse tangent in terms of the symbols used in the drawing. For instance, Equation 1.4 is read as “ equals the angle whose sine is ho/h.”

| (1.4)

| (1.5)

| (1.6)

The use of “” as an exponent in Equations 1.4, 1.5 and 1.6 does not mean “take the reciprocal.” For instance, does not equal ). Another way to express the inverse trigonometric functions is to use arc sin, arc cos, and arc tan instead of , , and . Example 5 illustrates the use of an inverse trigonometric function.

| Example 5 | | | Using Inverse Trigonometric Functions

A lakefront drops off gradually at an angle , as Figure 1-7 indicates. For safety reasons, it is necessary to know how deep the lake is at various distances from the shore. To provide some information about the depth, a lifeguard rows straight out from the shore a distance of 14.0 m and drops a weighted fishing line. By measuring the length of the line, the lifeguard determines the depth to be 2.25 m. (a) What is the value of ? (b) What would be the depth d of the lake at a distance of 22.0 m from the shore?

Figure 1-7 If the distance from the shore and the depth of the water at any one point are known, the angle can be found with the aid of trigonometry. Knowing the value of is useful, because then the depth d at another point can be determined.

Reasoning Near the shore, the lengths of the opposite and adjacent sides of the right triangle in Figure 1-7 are and , relative to the angle . Having made this identification, we can use the inverse tangent to find the angle in part (a). For part (b) the opposite and adjacent sides farther from the shore become and . With the value for obtained in part (a), the tangent function can be used to find the unknown depth. Considering the way in which the lake bottom drops off in Figure 1-7, we expect the unknown depth to be greater than 2.25 m.
Solution

a. | Using the inverse tangent given in Equation 1.6, we find that

b. | With , the tangent function given in Equation 1.3 can be used to find the unknown depth farther from the shore, where and . Since , it follows that

which is greater than 2.25 m, as expected.

The right triangle in Figure 1-5 provides the basis for defining the various trigonometric functions according to Equations 1.1, 1.2 and 1.3. These functions always involve an angle and two sides of the triangle. There is also a relationship among the lengths of the three sides of a right triangle. This relationship is known as the Pythagorean theorem and is used often in this text.

| Pythagorean Theorem
| The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides:

| (1.7)

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