How To Find A Missing Angle Of A Right Triangle

Right Triangle Trigonometry

Learning Objective(southward)

· 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 lx°.

· Solve practical problems using right triangle trigonometry.

Introduction

Suppose yous have to build a ramp and don't know how long it needs to be. You lot know certain angle measurements and side lengths, only yous need to notice the missing pieces of information.

In that location are six trigonometric functions, or ratios, that y'all tin use to compute what you lot don't know. Y'all will now learn how to use these six functions to solve correct triangle application problems.

Using the Pythagorean Theorem in Trigonometry Bug

There are several ways to make up one's mind the missing information in a right triangle. Ane of these means 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, equally shown below.

Topic2_1

If you lot know the length of any ii sides, then you tin can use the Pythagorean Theorem ( ) to find the length of the tertiary side. Once yous know all the side lengths, you lot tin can compute all of the trigonometric functions.

Example
Trouble	Find the values of and .
		You tin immediately find the tangent from the definition and the information in the diagram.
		To notice the value of the secant, you volition demand the length of the hypotenuse. Use the Pythagorean Theorem to find the length of the hypotenuse.
		Now calculate sec X using the definition of secant.
Reply

What is the value of ?

Show/Hibernate Reply

Wrong. You establish cos R instead csc R. Utilize the Pythagorean Theorem to find the contrary side length. Then carve up the hypotenuse past the opposite length. The right answer is .

Incorrect. Information technology looks like you used the incorrect bending and found . Remember that the two astute angles will requite you unlike trigonometric office values. Use the Pythagorean Theorem to detect the reverse side length. Divide the hypotenuse past the opposite length. The correct respond is .

Correct. You demand to know the opposite side length, so apply the Pythagorean Theorem: . This simplifies to . Solving this equation gives you . Using the definition of cosecant, .

Incorrect. You lot probably used the correct definition, , and used the Pythagorean Theorem to observe the opposite side length, r, but fix the equation incorrectly. It should be . The correct answer is .

Some problems may provide you lot with the values of 2 trigonometric ratios for 1 angle and ask yous to find the value of other ratios. Even so, y'all actually only demand to know the value of one trigonometric ratio to discover the value of any other trigonometric ratio for the same angle.

Example
Trouble	For acute angle A, . Find the values of and .
		Get-go you need to draw a right triangle in which . The tangent is the ratio of the contrary side to the next side. The simplest triangle you lot tin use that has that ratio is shown. It has an opposite side of length 2 and an next side of length 5. You could have used a triangle that has an opposite side of length 4 and an adjacent side of length 10. (You just need the ratio to reduce to).
		You can utilise the Pythagorean Theorem to detect the hypotenuse.
		Then use the definition of cosine to discover cos A.
		Now use the fact that sec A = i/cos A to find sec A.
Answer	,

Example
Problem	If bending 10 is an astute angle with , what is the value of ?
		In this right triangle, because , the ratio of the contrary side to the hypotenuse is . The simplest triangle we can utilize that has that ratio would be the triangle that has an reverse side of length 3 and a hypotenuse of length 4.
		Nosotros can utilise the Pythagorean Theorem to find the unknown leg length.
		You lot can find the cotangent using the definition. Or you can find the cotangent by offset finding tangent and then taking the reciprocal.
Answer

Solving Right Triangles

Determining all of the side lengths and angle measures of a right triangle is known as solving a correct triangle. Let's look at how to do this when you're given 1 side length and one acute angle measure. Once you learn how to solve a right triangle, you'll exist able to solve many existent 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.

Example
Problem	You need to build a ramp with the following dimensions. Solve the right triangle shown below. Use the approximations and , and give the lengths to the nearest 10th.
		Remember that the astute angles in a correct triangle are complementary, which means their sum is ninety°. Since , it follows that .
		You lot can utilise the definition of cosecant to detect c. Substitute the measure of the angle on the left side of the equation and use the triangle to set the ratio on the right. Solving the equation and rounding to the nearest tenth gives you lot .
		In a like way, yous can use the definition of tangent and the measure of the angle to find b. Solving the equation and rounding to the nearest 10th gives you .
Answer		The ramp needs to exist 11.7 feet long.

In the problem above, you lot were given the values of the trigonometric functions. In the adjacent trouble, y'all'll need to use the trigonometric role keys on your calculator to find those values.

Instance
Problem	Solve the right triangle shown below. Give the lengths to the nearest 10th.
		The acute angles are complementary, which means their sum is 90°. Since , it follows that .
		You can apply the definition of sine to find x. Use your calculator to find the value of and the triangle to set up up the ratio on the right. Solving the equation and rounding to the nearest tenth gives you .
		To find y, you tin either apply another trigonometric function (such as cosine) or you can utilize the Pythagorean Theorem. Solving the equation and rounding to the nearest tenth gives you .
Answer		Nosotros now know all three sides and all three angles. Their values are shown in the cartoon.

Sometimes you may be given enough information about a correct triangle to solve the triangle, merely that information may not include the measures of the acute angles. In this situation, you lot volition demand to utilise the inverse trigonometric function keys on your calculator to solve the triangle.

Example
Problem	Solve the correct triangle shown below, given that . Discover the exact side lengths and approximate the angles to the nearest degree.
		You are not given an bending measure, but you can use the definition of cotangent to find the value of north.
		Utilize the ratio you are given on the left side and the information from the triangle on the right side. Cross-multiply and solve for northward.
		Use the Pythagorean Theorem to notice the value of p.
		We can use the triangle to discover a value of the tangent and the inverse tangent fundamental on your calculator to find the angle that yields that value. Rounding to the nearest degree, is approximately 39°, . Subtract 39°, from ninety° to get .
Answer		We at present know all three sides and all three angles. Their values are shown in the drawing.

What is the value of ten to the nearest hundredth?

Topic2_12

A) 4.57

B) 1.97

C) 0.90

D) 0.22

Show/Hibernate Answer

A) 4.57

Incorrect. Y'all probably fix upward the ratio incorrectly, equating and . A right way to gear up the equation is . The right answer is one.97.

B) ane.97

Correct. Ane way to set up a correct equation is to use the definition of cosine. This will give you . The solution to this equation is:

. This rounds to one.97.

C) 0.90

Incorrect. You probably gear up the right equation, , and solved it correctly. Nevertheless, your calculator was non set to degrees. The correct answer is 1.97.

D) 0.22

Incorrect. Yous may have correctly gear up your equation equally , but then incorrectly solved it every bit . The right answer is i.97.

Special Angles

Equally 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 discover the exact values of these functions without a estimator. Allow'south meet how.

Suppose you lot had a right triangle with an acute bending 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 accept the same length, for case, 1 unit.

Topic2_13

Yous can determine the hypotenuse using the Pythagorean Theorem.

At present you lot have all the sides and angles in this right triangle.

Topic2_14

You can employ this triangle (which is sometimes chosen a 45° - 45° - 90° triangle) to find all of the trigonometric functions for 45°. I style to remember this triangle is to note that the hypotenuse is times the length of either leg.

Example
Trouble	Detect the values of the six trigonometric functions for 45° and rationalize denominators, if necessary.
		Use the definitions of sine, cosine and tangent. Notice that because the opposite and adjacent sides are equal, sine and cosine are equal.
		Use the reciprocal identities. Observe that because the opposite and adjacent sides are equal, cosecant and secant are equal.
Answer

You lot can construct another triangle that you can utilize to find all of the trigonometric functions for xxx° and sixty°. Start with an equilateral triangle with side lengths equal to 2 units. If you separate the equilateral triangle down the middle, you produce two triangles with 30°, threescore° and 90° angles. These two right triangles are congruent. They both have a hypotenuse of length two and a base of operations of length 1.

You lot can decide the height using the Pythagorean Theorem.

Here is the left half of the equilateral triangle turned on its side.

Topic2_16

You can use this triangle (which is sometimes called a xxx° - 60° - 90° triangle) to find all of the trigonometric functions for 30° and 60°. Note that the hypotenuse is twice as long every bit the shortest leg which is opposite the 30° angle, so that . The length of the longest leg which is reverse the 60 ° bending is times the length of the shorter leg.

Instance
Problem	Find the values of . Rationalize denominators, if necessary.
		Employ the definitions of sine, cosine and tangent. For each angle, be sure to employ the legs that are reverse and side by side to that angle. For case, is opposite to sixty°, just next to thirty°.
		Think that secant is the reciprocal of cosine and that cotangent is the reciprocal of tangent. Rationalize the denominators.
Answer

Y'all tin use the information from the 30° - 60° - 90° and 45° - 45° - 90° triangles to solve similar triangles without using a estimator.

Case
Problem	*What is the value of ten* in the triangle below?**
		Since the ii legs have the aforementioned length, the two acute angles must be equal, so they are each 45°.
		In a 45° - 45° - 90° triangle, the length of the hypotenuse is times the length of a leg. Y'all tin can use this relationship to find x. Remember to rationalize the denominator. Here is another way you solve this problem. Yous can use the definition of sine to find x.
Answer

You too could have solved the final problem using the Pythagorean Theorem, which would have produced the equation .

Example
Problem	Solve the right triangle shown below.
		The acute angles are complementary, and so . This is a xxx°- 60°- 90° triangle. We tin can at present use the trigonometric functions to find the lengths of the missing sides.
		Since nosotros know all the measures of the angles, we now need to find the lengths of the missing sides. To find c (the length of the hypotenuse), nosotros can use the sine office because we know that and we know the length of the opposite side.
		To observe a (the length of the side opposite angle A), we can use the tangent function because we know that and we know the length of the adjacent side.
Answer		We now know all three sides and all three angles. Their values are shown in the drawing.

If , what is the value of ?

A) two

Show/Hide Reply

A) 2

Incorrect. Refer to the 30°- 60°- xc° triangle. From in that location you can see that . It follows that . You lot may have used the wrong angle and institute , or used the correct angle merely found . You need to observe . The correct respond is .

Wrong. Refer to the 30°- 60°- ninety° triangle. From there y'all can meet that . It follows that . You may have mistakenly establish . Yous demand to discover . The right respond is .

Incorrect. You may have correctly found , only fabricated a mistake when rationalizing the denominator. The right respond is .

Correct. Refer to the 30°- lx°- 90° triangle. From there you tin encounter that . It follows that . So .

Using Trigonometry in Real-World Issues

There are situations in the real earth, such every bit building a ramp for a loading dock, in which you lot take a right triangle with sure data nigh the sides and angles, and you wish to find unknown measures of sides or angles. This is where understanding trigonometry can help you.

Instance
Problem	Ben and Emma are out flying a kite. Emma tin see that the kite string she is holding is making a seventy° bending with the footing. The kite is directly above Ben, who is standing 50 feet abroad. To the nearest foot, how many feet of string has Emma permit out?
		We want to detect the length of cord allow out. It is the hypotenuse of the right triangle shown.
		Since the l foot altitude measures the adjacent side to the 70° angle, you can use the cosine part to find 10.
		Solve the equation for x. Use a calculator to find a numerical value. The answer rounds to 146.
Respond	Emma has let out approximately 146 feet of string.

In the example above, you lot were given 1 side and an acute angle. In the side by side 1, you're given two sides and asked to detect an angle. Finding an angle will usually involve using an inverse trigonometric function. The Greek alphabetic character theta, θ, is commonly used to represent an unknown bending. In this case, θ represents the angle of tiptop.

Case
Problem	A wheelchair ramp is placed over a set of stairs so that i end is 2 feet off the ground. The other end is at a point that is a horizontal altitude of 28 feet away, as shown in the diagram. What is the angle of elevation to the nearest tenth of a degree?
		The bending of meridian is labeled in the diagram. The lengths given are the sides opposite and adjacent to this bending, and then you lot tin use the tangent function to find .
		Y'all want to find the measure of an bending that gives you a certain tangent value. This means that y'all need to observe the changed tangent. Retrieve that yous have to use the keys 2ND and TAN on your calculator. Look at the hundredths identify to round to the nearest 10th.
Answer	The angle of elevation is approximately iv.1°.

Remember that problems involving triangles with certain special angles can be solved without the use of a calculator.

Case
Problem	A contend is used to brand a triangular enclosure with the longest side equal to 30 anxiety, as shown below. What is the exact length of the side opposite the 60° angle?
		Call the unknown length x. Since you know the length of the hypotenuse, yous can use the sine part.
		This is a 30°- 60°- ninety° triangle. Therefore, you tin find the exact value of the trigonometric function without using a reckoner.
		Solve the equation for x.
Reply	The exact length of the side reverse the 60°angle is feet.

Sometimes the right triangle can be part of a bigger motion picture.

A guy wire is attached to a telephone pole 3 feet beneath the tiptop of the pole, as shown beneath. The guy wire is anchored xiv feet from the telephone pole and makes a 64° angle with the footing. How loftier up the pole is the guy wire attached? Round your answer to the nearest tenth of a human foot.

Bear witness/Hide Respond

Incorrect. You may have been confused as to which ratio corresponds to which trigonometric function. You need to solve the equation , where x represents the vertical distance from the base of the telephone pole upwards to where the guy wire is fastened. The correct respond is .

Correct. Let ten stand for the vertical altitude from the base of operations of the telephone pole upwards to where the guy wire is attached. And then . Solving this equation for x gives y'all .

Incorrect. Information technology looks similar yous prepare and solved the right equation to find the unknown length. Notwithstanding, y'all misread the problem. When you added the 3 you found the height of the entire pole. The correct answer is .

Wrong. Information technology looks like you set upwardly and solved an equation to detect the length of the wire (the hypotenuse of the triangle). You lot need to solve the equation , where x represents the vertical distance from the base of operations of the telephone pole up to where the guy wire is attached. The correct respond is .

Summary

There are many ways to notice the missing side lengths or angle measures in a correct triangle. Solving a right triangle can be accomplished past using the definitions of the trigonometric functions and the Pythagorean Theorem. This process is called solving a right triangle. Beingness able to solve a right triangle is useful in solving a diversity of real-world problems such as the structure of a wheelchair ramp.

Yous tin observe the exact values of the trigonometric functions for angles that measure out 30°, 45°, and 60°. Yous can find exact values for the sides in 30 ° , 45 ° , and 60 ° triangles if you remember that and . For other bending measures, it is necessary to use a calculator to find guess values of the trigonometric functions.