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Right Triangle Trigonometry Help

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By McGraw-Hill Professional
Updated on Sep 6, 2011

Introduction to Right Triangle Trigonometry

Using trigonometry to solve triangles is one of the oldest forms of mathematics. One of its most powerful uses is to measure distances—the height of a tree or building, the distance between earth and the moon, or the dimensions of a plot of land. The trigonometric ratios below are the same as before with the unit circle, only the labels are different. We will begin with right triangles.

In a right triangle, one angle measures 90° and the sum of the other angles is also 90°. The side opposite the 90° angle is the hypotenuse . The other sides are the legs . If we let θ represent one of the acute angles, then one of the legs is the side opposite θ , and the other side is adjacent to θ . See Figure 13.34.

Fig. 13.34

We can get the identity sin 2 θ + cos 2 θ = 1 from the Pythagorean Theorem. Opposite 2 + Adjacent 2 = Hypotenuse 2 Divide both sides by Hypotenuse 2.

sin 2 θ + cos 2 θ = 1

From this equation, we get two others, one from dividing both sides of the equation by sin 2 θ , and the other by dividing both sides by cos 2 θ.

Examples

• Find all six trigonometric ratios for θ.

Fig.13.35

• Find sin A , cos B , sec A , csc B , tan A , and cot B.

Fig. 13.36

The hypotenuse is 13, the side opposite ∠ A is 5, so sin A = 5/13. The side adjacent to ∠ B is 5, so cos B = 5/13. The other ratios are sec A = 13/12, csc B = 13/12, tan A = 5/12, and cot B = 5/12.

Co-Functions

The side opposite ∠ A is the side adjacent to ∠ B , and the side adjacent to ∠ A is opposite ∠ B . This is why sine and cosine, secant and cosecant, and tangent and cotangent are co-functions. Because ∠ A + ∠ B = 90°, we have ∠ B = 90° − ∠ A . These facts give us the following important relationships.

Solving a Triangle

To “solve a triangle” means to find all three angles and the lengths of all three sides. For now, we will solve right triangles. Later, after covering inverse trigonometric functions, we can solve other triangles. When solving right triangles, we will use the Pythagorean Theorem as well as the fact that the sum of the two acute angles is 90°. Except for the angles 30°, 45°, and 60°, we need a calculator. The calculator should be in degree mode. Also, there are probably no keys for secant, cosecant, and cotangent. You will need to use the reciprocal key, marked either or x −1 . The keys marked sin −1 , cos −1 , and tan −1 are used to evaluate the functions covered in the next section.

Examples

• Solve the triangle.

Fig. 13.37

The side opposite the angle 30° is 3, so . We know that . This gives us an equation to solve.

We could use trigonometry to find the third side, but it is usually easier to use the Pythagorean Theorem.

Angle of Elevation or Depression

In some applications of right triangles, we are given the angle of elevation or depression to an object. The angle of elevation is the measure of upward rotation. The angle of depression is the measure of the downward rotation. See Figure 13.38.

Fig. 13.38

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