Right Triangle Trigonometry Help (page 2)
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.
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.
- Find all six trigonometric ratios for θ.
- Find sin A , cos B , sec A , csc B , tan A , and cot B.
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.
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.
- Solve the triangle.
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.
- A person is standing 300 feet from the base of a five-story building. He estimates that the angle of elevation to the top of the building is 63°. Approximately how tall is the building?
We need to find b in the following triangle.
We could use either of the ratios that use the opposite and adjacent sides, tangent (opposite/adjacent) and cotangent (adjacent/opposite). We will use tangent.
This gives us the equation tan 63° = b /300. When we solve for b , we have b = 300 tan 63° ≈ (300)1.9626 ≈ 588.78. The building is about 589 feet tall.
- A guy wire is 60 feet from the base of a tower. The angle of elevation from the top of the tower along the wire is 73°. How long is the wire?
We need to find c in the following triangle.
We could use either cosine (adjacent/hypotenuse) or secant (hypotenuse/adjacent). Using cosine, we have cos 73° = 60/ c . Solving this equation for c gives us c = 60/cos 73° ≈ 60/0.2924 ≈ 205. The wire is about 205 feet long.
Right Triangle Trigonometry Practice Problems
Find all six trigonometric ratios for θ.
Solve the triangle.
A plane is flying at an altitude of 5000 feet. The angle of elevation to the plane from a car traveling on a highway is about 38.7°. How far apart are the plane and car?
We could use any of the ratios involving the hypotenuse. We will use cosine: cos 60° = a /4. Since cos 60° = 1/2, we have 1/2 = a /4. Solving for a gives us a = 2.
We need to find c in the following triangle.
Practice problems for this concept can be found at: Trigonometry Practice Test.
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