Question

finding angle measures
note: \\( \sin ^ { - 1 } x \\) is read as the inverse sine of x.
given the trigonometric value of an angle, use the corresponding inverse trigonometric function to find the measure of the angle.
find the measure of \\( \theta \\). round to the nearest tenth of a degree.
7.
8.
9.
10.
11.
12.

Explanation:

Step1: Identify the trigonometric ratio

For problem 7:
We know that \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, adjacent side \(= 18\) and hypotenuse \(=23\). So, \(\cos\theta=\frac{18}{23}\)

Step2: Use the inverse cosine function

\(\theta=\cos^{-1}(\frac{18}{23})\)
Using a calculator, \(\cos^{-1}(\frac{18}{23})\approx38.9^{\circ}\)

For problem 8:
We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, opposite side \(=5\) and adjacent side \(=21\). So, \(\tan\theta=\frac{5}{21}\)

Step3: Use the inverse tangent function

\(\theta = \tan^{-1}(\frac{5}{21})\)
Using a calculator, \(\tan^{-1}(\frac{5}{21})\approx13.3^{\circ}\)

For problem 9:
First, find the hypotenuse using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 17\) and \(b=18\). \(c=\sqrt{17^{2}+18^{2}}=\sqrt{289 + 324}=\sqrt{613}\approx24.8\)
We know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, opposite side \(=17\) and hypotenuse \(\approx24.8\). So, \(\sin\theta=\frac{17}{24.8}\)

Step4: Use the inverse sine function

\(\theta=\sin^{-1}(\frac{17}{24.8})\)
Using a calculator, \(\sin^{-1}(\frac{17}{24.8})\approx43.3^{\circ}\)

For problem 10:
We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, opposite side \(=39\) and adjacent side \(=20\). So, \(\tan\theta=\frac{39}{20}\)

Step5: Use the inverse tangent function

\(\theta=\tan^{-1}(\frac{39}{20})\)
Using a calculator, \(\tan^{-1}(\frac{39}{20})\approx62.9^{\circ}\)

For problem 11:
We know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, opposite side \(=29\) and hypotenuse \(=52\). So, \(\sin\theta=\frac{29}{52}\)

Step6: Use the inverse sine function

\(\theta=\sin^{-1}(\frac{29}{52})\)
Using a calculator, \(\sin^{-1}(\frac{29}{52})\approx34.0^{\circ}\)

For problem 12:
First, find the adjacent side using the Pythagorean theorem \(a=\sqrt{c^{2}-b^{2}}\), where \(c = 19\) and \(b = 12\). \(a=\sqrt{19^{2}-12^{2}}=\sqrt{361-144}=\sqrt{217}\approx14.7\)
We know that \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, adjacent side \(\approx14.7\) and hypotenuse \(=19\). So, \(\cos\theta=\frac{14.7}{19}\)

Step7: Use the inverse cosine function

\(\theta=\cos^{-1}(\frac{14.7}{19})\)
Using a calculator, \(\cos^{-1}(\frac{14.7}{19})\approx38.3^{\circ}\)

Answer:

1. \(38.9^{\circ}\)
2. \(13.3^{\circ}\)
3. \(43.3^{\circ}\)
4. \(62.9^{\circ}\)
5. \(34.0^{\circ}\)
6. \(38.3^{\circ}\)