A right triangle has legs that measure 2 units and
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,
Example A
Find the measure of angle

sinA=0.8336 
tanA=1.3527 
cosA=0.2785
Solution: Using the calculator we get the following:

sin−1(0.8336)≈56∘ 
tan−1(1.3527)≈54∘ 
cos−1(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
Recall that in a right triangle, the acute angles are always complementary, so
Example C
Solve the right triangle shown below. Round all answers to the nearest tenth.
Solution: We can solve for either angle
Now we can find
Method 1: We can using trigonometry and the cosine ratio:
Method 2: We can subtract
Either method is valid, but be careful with Method 2 because a miscalculation of angle
Concept Problem Revisit
First, let's find the hypotenuse, then we can solve for either angle.
One of the acute angles will have a sine of
Now we can find
Guided Practice
1. Find the measure of angle
a.
b.
c.
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.
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{38^217^2} \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.
 \begin{align*}\tan B=0.9523\end{align*}
 \begin{align*}\sin B=0.8659\end{align*}
 \begin{align*}\cos B=0.1568\end{align*}
 \begin{align*}\sin B=0.2234\end{align*}
 \begin{align*}\cos B=0.4855\end{align*}
 \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.