Lesson Explainer: Right Triangle Trigonometry: Solving for an Angle | Nagwa Lesson Explainer: Right Triangle Trigonometry: Solving for an Angle | Nagwa

Lesson Explainer: Right Triangle Trigonometry: Solving for an Angle Mathematics

In this explainer, we will learn how to find a missing angle in a right triangle using the appropriate inverse trigonometric function given two side lengths.

Recall that, when working with right triangles, we can use either the Pythagorean theorem or the trigonometric ratios to find unknown sides or angles. When we have two known sides and need to find a third side, we use the Pythagorean theorem. However, if we have a right triangle with one known side and one known angle, we can find an unknown side using the trigonometric ratios sine, cosine, and tangent. Let’s recap these below.

Definition: Trigonometric Ratios

For a right triangle with a nonright angle 𝜃, hypotenuse H, the side opposite to 𝜃, O, and the side adjacent to 𝜃, A, as seen in the diagram below, the following trigonometric ratios hold true: sinOHcosAHtanOA𝜃=,𝜃=,𝜃=.

For example, if we are given the measures of an angle, 𝜃, and the side adjacent to it, A, and wanted to find the opposite side, O, we could rearrange the trigonometric ratio tanOA𝜃= to get OAtan=𝜃.

In addition to finding an unknown side, we can use the trigonometric ratios to find an unknown angle when two sides are known. To do this we use the trigonometric inverse of sine, cosine, and tangent in order to rearrange to make the angle the subject of the equation.

Definition: Trigonometric Inverse

For an angle 𝜃 the trigonometric inverse for sine, cosine, and tangent can be written as follows:

  • For sinOH𝜃=, the inverse can be written as 𝜃=sinOH.
  • For cosAH𝜃=, the inverse can be written as 𝜃=cosAH.
  • For tanOA𝜃=, the inverse can be written as 𝜃=tanOA.

As with solving for an unknown side, when solving for an unknown angle, we start by labeling the sides of the triangle in correspondence with the angle we are trying to find. Recall that, for an angle 𝜃 in a right triangle, we label the hypotenuse as H, the side opposite the angle 𝜃 as O, and the side adjacent to the angle 𝜃 as A. Once we have labeled the sides, we can then determine which trigonometric ratio to use and substitute into it accordingly. We then use the trigonometric inverse to solve for the angle.

We will consider how to find an unknown angle given the diagram in the first example.

Example 1: Using the Inverse of a Sine Function to Find an Unknown Angle

In the given figure, find the measure of angle 𝜃, in degrees, to two decimal places.

Answer

In this question we are given a right triangle with two known sides and an unknown angle. Since we are required to find the missing angle, we need to use right angle trigonometry.

When solving problems involving right angle trigonometry, we need to label the figure in order to determine which trigonometric ratio to use. We label according to the angle we know or are trying to find, which, in this case, is 𝜃. We are given the length of the side opposite the right angle, which is the hypotenuse, and the length of the side opposite the angle 𝜃. We label these as H and O accordingly, as seen below.

We can see from the figure that Ocm=5 and Hcm=11. Therefore, we need to use the trigonometric ratio that contains O and H. This is sinOH𝜃=.

Substituting O=5 and H=11, we get sin𝜃=511.

Now that we have substituted into the sine ratio, we need to solve for the unknown angle 𝜃. To do this, we rearrange the sine ratio by using the trigonometric inverse sin. Doing so, we get sinsin𝜃=511𝜃=511.

And then, solving using a calculator, we get 𝜃=27.03569.

Therefore, the measure of angle 𝜃 in degrees correct to 2 decimal places is 27.04.

In the next example, we will again find an unknown angle when given a figure but are told which angles to find in angle notation.

Example 2: Using the Inverse Tangent Function to Find an Unknown Angle

For the given figure, find the measure of 𝐵𝐴𝐶, in degrees, to two decimal places.

Answer

In this question, we are asked to find the measure of 𝐵𝐴𝐶 and, from the figure, have been given the lengths of two sides in a right triangle. As such, we need to use right angle trigonometry to find the unknown angle.

When solving problems involving right angle trigonometry, we label the angle we are trying to find 𝜃 and then the corresponding known sides. We are given the side opposite to 𝐵𝐴𝐶 and the side adjacent to 𝐵𝐴𝐶. Therefore, we label 𝐵𝐴𝐶 as 𝜃, the opposite side as O, and the adjacent side as A. This is shown in the diagram below.

We can see from the figure that Ocm=7 and Acm=5. Therefore, we need to use the trigonometric ratio that contains A and O. This is tanOA𝜃=.

Substituting O=7 and A=5, we get tan𝜃=75.

Now that we have substituted into the tangent ratio, we need to solve for the unknown angle 𝜃. To do this, we rearrange the tangent ratio by using the trigonometric inverse tan. Doing so, we get tantan𝜃=75𝜃=75.

And, solving using a calculator, we get 𝜃=54.4623.

Therefore, the measure of 𝐵𝐴𝐶 is 54.46 to 2 decimal places.

In the following example, we are asked to find the measures of two unknown angles instead of just one.

Example 3: Finding the Unknown Angles in a Right Triangle Using Trigonometry

For the given figure, find the measures of 𝐴𝐵𝐶 and 𝐴𝐶𝐵, in degrees, to two decimal places.

Answer

In this question, we are asked to find two unknown angles in a right triangle while given two sides in the figure. As such, we need to use right angle trigonometry to solve for the unknown angles.

When solving problems involving right angle trigonometry, we label the angle we are trying to find 𝜃 and then the corresponding known sides. As we are trying to find two unknown angles, we will start by trying to find angle 𝐴𝐵𝐶. We are given the length of the side opposite the right angle, the hypotenuse, and the length of the side adjacent to the angle 𝐴𝐵𝐶. We therefore label 𝐴𝐵𝐶 as 𝜃, the hypotenuse as H, and the adjacent as A, as shown below.

We can see from the figure that Acm=4 and Hcm=9. Therefore, we need to use the trigonometric ratio that contains A and H. This is cosAH𝜃=.

Substituting A=4 and H=9, we get cos𝜃=49.

Now that we have substituted into the cosine ratio, we need to solve for the unknown angle 𝜃. To do this, we rearrange the cosine ratio by using the trigonometric inverse cos. Doing so, we get coscos𝜃=49𝜃=49.

And, solving using a calculator, we get 𝜃=63.6122.

So, the measure of 𝐴𝐵𝐶 is 63.61 correct to 2 decimal places.

To find 𝐴𝐶𝐵 we use the property that the sum of the interior angles in a triangle is 180, since we know the other two angles. This gives us 𝑚𝐴𝐶𝐵+63.61+90=180𝑚𝐴𝐶𝐵=1809063.61𝑚𝐴𝐶𝐵=26.392.correcttodecimalplaces

Therefore, 𝐴𝐵𝐶 is 63.61 and 𝐴𝐶𝐵 is 26.39 correct to 2 decimal places.

In the following example, we are given the measures of two sides and are required to find an unknown side and unknown angles without a figure being provided.

Example 4: Using Trigonometry to Solve Right Triangles with Angles in Degrees

𝐴𝐵𝐶 is a right triangle at 𝐵, where 𝐵𝐶=10cm and 𝐴𝐶=18cm. Find the length 𝐴𝐵, giving the answer to the nearest centimetre, and the measure of angles 𝐴 and 𝐶, giving the answer to the nearest degree.

Answer

In this question, we are required to find an unknown side and two unknown angles in a right triangle. To do this, it is helpful to first draw a diagram to represent the information given. We will start by drawing a right triangle 𝐴𝐵𝐶 and labeling the known sides, 𝐵𝐶=10cm and 𝐴𝐶=18cm, as shown below.

We can solve this question using right triangle trigonometry. As we are given two sides, we will start by finding the unknown angles. Let’s start by finding the angle 𝐴.

To do this, it helps to label our diagram with angle 𝐴 as our unknown angle 𝜃, 𝐴𝐶 as the hypotenuse H, and 𝐵𝐶 as the opposite side O. The diagram below shows how 𝜃, H, and O have been labeled.

From the diagram we can see that Ocm=10 and Hcm=18. Therefore, we need to use the trigonometric ratio that contains O and H, which is sinOH𝜃=.

Substituting O=10 and H=18, we get sin𝜃=1018.

Now that we have substituted into the sine ratio, we need to solve for the unknown angle 𝜃. To do this we rearrange the sine ratio by using the trigonometric inverse sin. Doing so, we get sinsin𝜃=1018𝜃=1018.

And, solving using a calculator, we get 𝜃=33.749.

Therefore, 𝑚𝐴 is 34 to the nearest degree.

Next, to find angle 𝐶, we use the property that the sum of the interior angles in a triangle is 180. Since we know the other two angles, this gives us 𝑚𝐶+33.749+90=180𝑚𝐶=1809033.749𝑚𝐶=56.251.

Therefore, 𝑚𝐶 is 56 to the nearest degree.

Lastly, we need to find side 𝐴𝐵. Using right triangle trigonometry, we can use one of the known sides and one of the known angles to do this. As we have two known sides and two known angles, we can choose which trigonometric ratio to use. As we found 𝑚𝐴 first, we will use this angle as 𝜃. We will use the opposite side, O, leaving us with the side we are trying to find as the adjacent side A. We can see this below.

As we know, Ocm=10, 𝜃=33.749, and we are trying to find A. We need to use the trigonometric ratio that contains O and A. This is tanOA𝜃=.

Substituting O=10 and 𝜃=33.749, we get tanA(33.749)=10.

Rearranging for A, we get AtanAtan(33.749)=10=10(33.749).

And, solving using a calculator, we get Acm=14.9666.

Therefore, the length of 𝐴𝐵 is 15 cm correct to the nearest cm.

So, the length of 𝐴𝐵 to the nearest centimetre is 15 cm, the measure of angle 𝐴 to the nearest degree is 34, and the measure of angle 𝐶 to the nearest degree is 56, as required.

In the next example, we will consider how to find an unknown angle in the context of a real-life problem.

Example 5: Solving Real Life Problems with Trigonometry

A 5 m ladder is leaning against a vertical wall such that its base is 2 m from the wall. Work out the angle between the ladder and the floor, giving your answer to two decimal places.

Answer

In this question we are given information about the length of a ladder and how far the base is from a wall. To solve this, it is helpful to first draw a diagram to illustrate the information given.

As the question asks for the angle between the ladder and the floor and we know the length of two sides, we use right angle trigonometry to find the unknown angle. We can label the unknown angle as 𝜃, the side opposite the right angle, the hypotenuse, as H, and the side adjacent to the angle as A.

From the diagram, we can see that Am=2 and Hm=5. Therefore, we need to use the trigonometric ratio that contains A and H. This is cosAH𝜃=.

Substituting A=2 and H=5, we get cos𝜃=25.

Now that we have substituted into the cosine ratio, we need to solve for the unknown angle 𝜃. To do this, we rearrange the cosine ratio by using the trigonometric inverse cos. Doing so, we get coscos𝜃=25𝜃=25.

And, solving using a calculator, we get 𝜃=66.4218.

Therefore, the angle between the ladder and the floor is 66.42 to 2 decimal places.

In the last example, we will discuss how to find an unknown angle in a real-life problem.

Example 6: Finding Unknown Angles in Real-Life Problems Using Trigonometry

A car is going down a ramp that is 10 metres high and 71 metres long. Find the angle between the ramp and the horizontal, giving the answer to the nearest second.

Answer

In this question we are required to find the angle between the ramp and the horizontal and are told that the ramp is 10 metres high and 71 metres long. We can draw a right triangle, with height 10 metres and hypotenuse 71 metres to illustrate this.

Next, we want to identify the unknown we are looking for, which is the angle between the ramp and the horizontal. We will label this as 𝜃 on our diagram, as follows.

As we have two known sides and one unknown angle in a right triangle, we can use right triangle trigonometry to find the unknown angle. Since the known sides are the height, which is the side opposite the angle, and the ramp, which is the hypotenuse, we can label the height and ramp as O and H respectively.

We can see from the figure that Om=10 and Hm=71. Therefore, we need to use the trigonometric ratio that contains O and H. This is sinOH𝜃=.

Substituting O=10 and H=71, we get sin𝜃=1071.

Now that we have substituted into the sine ratio, we need to solve for the unknown angle 𝜃. To do this we rearrange the sine ratio by using the trigonometric inverse sin. Doing so, we get sinsin𝜃=1071𝜃=1071.

And, solving using a calculator, we get 𝜃=8.09674.

As the question requires the answer to the nearest second, we need to calculate the number of minutes and seconds given rather than the decimal of a degree.

Since we have 0.09674 degrees, then, by multiplying by 60, we get 5.805 minutes. Again, since we have 0.805 minutes, by multiplying by 60, we get 48.3 seconds, which is 48 to the nearest second. This gives us an answer of 8 degrees, 5 minutes, and 48 seconds to the nearest second, or 8548.

In this explainer we have learned how to solve for an angle when using right triangle trigonometry. We have done this using the trigonometric inverses of sine, cosine, and tangent. Let’s recap the key points.

Key Points

  • To solve for an unknown angle in a right triangle, where two sides are known, we use right triangle trigonometry. This involves using one of the following trigonometric ratios:
    • sinOH𝜃=,
    • cosAH𝜃=,
    • tanOA𝜃=,
    where we have a nonright angle 𝜃, a hypotenuse of length H, the side opposite to 𝜃 of length O, and the side adjacent to 𝜃 of length A, as seen below.
  • For the sine, cosine, and tangent of an angle 𝜃, we use the trigonometric inverse to solve for 𝜃 as follows:
    • For sinOH𝜃=, the inverse can be written as 𝜃=sinOH.
    • For cosAH𝜃=, the inverse can be written as 𝜃=cosAH.
    • For tanOA𝜃=, the inverse can be written as 𝜃=tanOA.
  • When solving problems, it is helpful to draw a diagram if one is not given and then to label the sides and unknown angle in order to correctly identify which trigonometric ratio to use.

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