<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

Basic Inverse Trigonometric Functions

Apply inverse trig ratios to determine the measure of unknown angles

Atoms Practice
%
Progress
Practice
Progress
%
Practice Now
Turn In
Inverse Trig Functions and Solving Right Triangles

A right triangle has legs that measure 2 units and 23 units. What are the measures of the triangle's acute angles?

Guidance

In the previous concept we used the trigonometric functions sine, cosine and tangent to find the ratio of particular sides in a right triangle given an angle. In this concept we will use the inverses of these functions, sin1, cos1 and tan1, to find the angle measure when the ratio of the side lengths is known. When we type sin30 into our calculator, the calculator goes to a table and finds the trig ratio associated with 30, which is 12. When we use an inverse function we tell the calculator to look up the ratio and give us the angle measure. For example: sin1(12)=30. On your calculator you would press 2NDSIN to get SIN1( and then type in 12, close the parenthesis and press ENTER. Your calculator screen should read SIN1(12) when you press ENTER.

Example A

Find the measure of angle A associated with the following ratios. Round answers to the nearest degree.

  1. sinA=0.8336
  2. tanA=1.3527
  3. cosA=0.2785

Solution: Using the calculator we get the following:

  1. sin1(0.8336)56
  2. tan1(1.3527)54
  3. cos1(0.2785)74

Example B

Find the measures of the unknown angles in the triangle shown. Round answers to the nearest degree.

Solution: We can solve for either x or y first. If we choose to solve for x first, the 23 is opposite and 31 is adjacent so we will use the tangent ratio.

x=tan1(2331)37.

Recall that in a right triangle, the acute angles are always complementary, so 9037=53, so y=53. We can also use the side lengths an a trig ratio to solve for y:

y=tan1(3123)53.

Example C

Solve the right triangle shown below. Round all answers to the nearest tenth.

Solution: We can solve for either angle A or angle B first. If we choose to solve for angle B first, then 8 is the hypotenuse and 5 is the opposite side length so we will use the sine ratio.

sinBmB=58=sin1(58)38.7

Now we can find A two different ways.

Method 1: We can using trigonometry and the cosine ratio: cosAmA=58=cos1(58)51.3

Method 2: We can subtract mB from 90: 9038.7=51.3 since the acute angles in a right triangle are always complimentary.

Either method is valid, but be careful with Method 2 because a miscalculation of angle B would make the measure you get for angle A incorrect as well.

Concept Problem Revisit

First, let's find the hypotenuse, then we can solve for either angle.

22+(23)=c24+12=c216=c2c=4

One of the acute angles will have a sine of 24=12.

sinAmA=12=sin112=30

Now we can find B by subtracting mA from 90: 9030=60 since the acute angles in a right triangle are always complimentary.

Guided Practice

1. Find the measure of angle A to the nearest degree given the trigonometric ratios.

a. sinA=0.2894

b. tanA=2.1432

c. cosA=0.8911

2. Find the measures of the unknown angles in the triangle shown. Round answers to the nearest degree.

3. Solve the triangle. Round side lengths to the nearest tenth and angles to the nearest degree.

Answers

1. a. sin1(0.2894)17

b. \begin{align*}\tan^{-1} (2.1432) \approx 65^\circ\end{align*}

c. \begin{align*}\cos^{-1} (0.8911) \approx 27^\circ\end{align*}

2. \begin{align*}x=\cos^{-1} \left(\frac{13}{20} \right) \approx 49^\circ; \quad y=\sin^{-1} \left(\frac{13}{20} \right) \approx 41^\circ\end{align*}

3. \begin{align*}m \angle A=\cos^{-1} \left(\frac{17}{38} \right) \approx 63^\circ; \quad m \angle B=\sin^{-1} \left(\frac{17}{38} \right) \approx 27^\circ; \quad a=\sqrt{38^2-17^2} \approx 34.0\end{align*}

Practice

Use your calculator to find the measure of angle \begin{align*}B\end{align*}. Round answers to the nearest degree.

  1. \begin{align*}\tan B=0.9523\end{align*}
  2. \begin{align*}\sin B=0.8659\end{align*}
  3. \begin{align*}\cos B=0.1568\end{align*}
  4. \begin{align*}\sin B=0.2234\end{align*}
  5. \begin{align*}\cos B=0.4855\end{align*}
  6. \begin{align*}\tan B=0.3649\end{align*}

Find the measures of the unknown acute angles. Round measures to the nearest degree.

Solve the following right triangles. Round angle measures to the nearest degree and side lengths to the nearest tenth.

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Basic Inverse Trigonometric Functions.
Please wait...
Please wait...