Right Triangle Trigonometry Learning Objective(s) · Use the Pythagorean Theorem to find the missing lengths of the sides of a right triangle. · Find the missing lengths and angles of a right triangle. · Find the exact trigonometric function values for angles that measure 30°, 45°, and 60°. · Solve applied problems using right triangle trigonometry. Introduction Suppose you have to build a ramp and don’t know how long it needs to be. You know certain angle measurements and side lengths, but you need to find the missing pieces of information. There are six trigonometric functions, or ratios, that you can use to compute what you don’t know. You will now learn how to use these six functions to solve right triangle application problems. Using the Pythagorean Theorem in Trigonometry Problems There are several ways to determine the missing information in a right triangle. One of these ways is the Pythagorean Theorem, which states that. Suppose you have a right triangle in which a and b are the lengths of the legs, and c is the length of the hypotenuse, as shown below. If you know the length of any two sides, then you can use the Pythagorean Theorem () to find the length of the third side. Once you know all the side lengths, you can compute all of the trigonometric functions.
Some problems may provide you with the values of two trigonometric ratios for one angle and ask you to find the value of other ratios. However, you really only need to know the value of one trigonometric ratio to find the value of any other trigonometric ratio for the same angle.
Solving Right Triangles Determining all of the side lengths and angle measures of a right triangle is known as solving a right triangle. Let’s look at how to do this when you’re given one side length and one acute angle measure. Once you learn how to solve a right triangle, you’ll be able to solve many real world applications – such as the ramp problem at the beginning of this lesson – and the only tools you’ll need are the definitions of the trigonometric functions, the Pythagorean Theorem, and a calculator.
In the problem above, you were given the values of the trigonometric functions. In the next problem, you’ll need to use the trigonometric function keys on your calculator to find those values.
Sometimes you may be given enough information about a right triangle to solve the triangle, but that information may not include the measures of the acute angles. In this situation, you will need to use the inverse trigonometric function keys on your calculator to solve the triangle.
What is the value of x to the nearest hundredth? A) 4.57 B) 1.97 C) 0.90 D) 0.22 Special Angles As a general rule, you need to use a calculator to find the values of the trigonometric functions for any particular angle measure. However, angles that measure 30°, 45°, and 60°—which you will see in many problems and applications—are special. You can find the exact values of these functions without a calculator. Let’s see how. Suppose you had a right triangle with an acute angle that measured 45°. Since the acute angles are complementary, the other one must also measure 45°. Because the two acute angles are equal, the legs must have the same length, for example, 1 unit. You can determine the hypotenuse using the Pythagorean Theorem. Now you have all the sides and angles in this right triangle. You can use this triangle (which is sometimes called a 45° - 45° - 90° triangle) to find all of the trigonometric functions for 45°. One way to remember this triangle is to note that the hypotenuse is times the length of either leg.
You can construct another triangle that you can use to find all of the trigonometric functions for 30° and 60°. Start with an equilateral triangle with side lengths equal to 2 units. If you split the equilateral triangle down the middle, you produce two triangles with 30°, 60° and 90° angles. These two right triangles are congruent. They both have a hypotenuse of length 2 and a base of length 1. You can determine the height using the Pythagorean Theorem. Here is the left half of the equilateral triangle turned on its side. You can use this triangle (which is sometimes called a 30° - 60° - 90° triangle) to find all of the trigonometric functions for 30° and 60°. Note that the hypotenuse is twice as long as the shortest leg which is opposite the 30° angle, so that . The length of the longest leg which is opposite the 60° angle is times the length of the shorter leg.
You can use the information from the 30° - 60° - 90° and 45° - 45° - 90° triangles to solve similar triangles without using a calculator.
You also could have solved the last problem using the Pythagorean Theorem, which would have produced the equation .
Using Trigonometry in Real-World Problems There are situations in the real world, such as building a ramp for a loading dock, in which you have a right triangle with certain information about the sides and angles, and you wish to find unknown measures of sides or angles. This is where understanding trigonometry can help you.
In the example above, you were given one side and an acute angle. In the next one, you’re given two sides and asked to find an angle. Finding an angle will usually involve using an inverse trigonometric function. The Greek letter theta, θ, is commonly used to represent an unknown angle. In this example, θ represents the angle of elevation.
Remember that problems involving triangles with certain special angles can be solved without the use of a calculator.
Sometimes the right triangle can be part of a bigger picture. A guy wire is attached to a telephone pole 3 feet below the top of the pole, as shown below. The guy wire is anchored 14 feet from the telephone pole and makes a 64° angle with the ground. How high up the pole is the guy wire attached? Round your answer to the nearest tenth of a foot. A) B) C) D) Summary There are many ways to find the missing side lengths or angle measures in a right triangle. Solving a right triangle can be accomplished by using the definitions of the trigonometric functions and the Pythagorean Theorem. This process is called solving a right triangle. Being able to solve a right triangle is useful in solving a variety of real-world problems such as the construction of a wheelchair ramp. You can find the exact values of the trigonometric functions for angles that measure 30°, 45°, and 60°. You can find exact values for the sides in 30°, 45°, and 60° triangles if you remember that and . For other angle measures, it is necessary to use a calculator to find approximate values of the trigonometric functions. How do u find the value of x in a triangle?Solving for X in a Right Triangle
Subtract the sum of the two angles from 180 degrees. The sum of all the angles of a triangle always equals 180 degrees. Write down the difference you found when subtracting the sum of the two angles from 180 degrees. This is the value of X.
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