Question

directions: give each trig ratio as a fraction in simplest form.
1.
directions: solve for x. round to the nearest tenth.
2.
3.
4.
5.
6.
7.
8.
9.

Explanation:

Step1: Find side DF in right - triangle DEF

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). In right - triangle \(DEF\) with \(DE = 29\) and \(EF = 20\), let \(DF=x\). Then \(x=\sqrt{DE^{2}-EF^{2}}=\sqrt{29^{2}-20^{2}}=\sqrt{(29 + 20)(29 - 20)}=\sqrt{49\times9}=\sqrt{441}=21\).

Step2: Recall trigonometric ratios

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)
For \(\angle D\):

• \(\sin D=\frac{EF}{DE}=\frac{20}{29}\)
• \(\cos D=\frac{DF}{DE}=\frac{21}{29}\)
• \(\tan D=\frac{EF}{DF}=\frac{20}{21}\)

For \(\angle E\):

• \(\sin E=\frac{DF}{DE}=\frac{21}{29}\)
• \(\cos E=\frac{EF}{DE}=\frac{20}{29}\)
• \(\tan E=\frac{DF}{EF}=\frac{21}{20}\)

Step3: Solve for \(x\) in right - triangle problems

Problem 2

We know \(\tan62^{\circ}=\frac{x}{25}\), so \(x = 25\times\tan62^{\circ}\approx25\times1.8807\approx47.0\)

Problem 3

We know \(\tan26^{\circ}=\frac{11}{x}\), so \(x=\frac{11}{\tan26^{\circ}}\approx\frac{11}{0.4877}\approx22.6\)

Problem 4

We know \(\sin48^{\circ}=\frac{x}{32}\), so \(x = 32\times\sin48^{\circ}\approx32\times0.7431\approx23.8\)

Problem 5

We know \(\sin12^{\circ}=\frac{x}{29}\), so \(x = 29\times\sin12^{\circ}\approx29\times0.2079\approx6.0\)

Problem 6

We know \(\cos x=\frac{15}{14}\) (this is incorrect as \(\cos x\in[- 1,1]\), assume it's \(\cos x=\frac{14}{15}\), then \(x=\cos^{-1}(\frac{14}{15})\approx21.0^{\circ}\))

Problem 7

We know \(\sin x=\frac{19}{23}\), then \(x=\sin^{-1}(\frac{19}{23})\approx55.8^{\circ}\)

Problem 8

We know \(\sin x=\frac{9}{17}\), then \(x=\sin^{-1}(\frac{9}{17})\approx32.1^{\circ}\)

Problem 9

We know \(\tan x=\frac{43}{45}\), then \(x=\tan^{-1}(\frac{43}{45})\approx43.6^{\circ}\)

Answer:

\(\sin D=\frac{20}{29}\), \(\cos D=\frac{21}{29}\), \(\tan D=\frac{20}{21}\), \(\sin E=\frac{21}{29}\), \(\cos E=\frac{20}{29}\), \(\tan E=\frac{21}{20}\)
\(x\approx47.0\) (for problem 2), \(x\approx22.6\) (for problem 3), \(x\approx23.8\) (for problem 4), \(x\approx6.0\) (for problem 5), \(x\approx21.0^{\circ}\) (for problem 6), \(x\approx55.8^{\circ}\) (for problem 7), \(x\approx32.1^{\circ}\) (for problem 8), \(x\approx43.6^{\circ}\) (for problem 9)