Trigonometric Ratios Worksheets - Math Monks - Free Printable

Let's solve each of these right triangle problems using trigonometric ratios (sine, cosine, and tangent). We'll use the following definitions for a right triangle:

- $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
- $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
- $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

We’ll identify which sides are given and which side we’re solving for (marked as $x$), then apply the correct trig ratio.

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Problem 1

Given:
- Angle = 46°
- Opposite side = 7
- Adjacent side = $x$
- Right angle at bottom right

We need to find $x$, adjacent to 46°, opposite is 7.

Use tangent:
$$
\tan(46^\circ) = \frac{7}{x}
\Rightarrow x = \frac{7}{\tan(46^\circ)}
$$
$$
\tan(46^\circ) \approx 1.0355
\Rightarrow x \approx \frac{7}{1.0355} \approx 6.76 \approx \boxed{6.8}
$$

✔ Answer: $x = 6.8$

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Problem 2

Given:
- Angle = 47°
- Hypotenuse = 3
- Opposite side = $x$

Use sine:
$$
\sin(47^\circ) = \frac{x}{3}
\Rightarrow x = 3 \cdot \sin(47^\circ)
$$
$$
\sin(47^\circ) \approx 0.7314
\Rightarrow x \approx 3 \cdot 0.7314 = 2.1942 \approx \boxed{2.2}
$$

✔ Answer: $x = 2.2$

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Problem 3

Given:
- Angle = 50.1°
- Adjacent side = 5
- Opposite side = $x$

Use tangent:
$$
\tan(50.1^\circ) = \frac{x}{5}
\Rightarrow x = 5 \cdot \tan(50.1^\circ)
$$
$$
\tan(50.1^\circ) \approx 1.1918
\Rightarrow x \approx 5 \cdot 1.1918 = 5.959 \approx \boxed{6.0}
$$

✔ Answer: $x = 6.0$

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Problem 4

Given:
- Angle = 60°
- Hypotenuse = 11
- Opposite side = $x$

Use sine:
$$
\sin(60^\circ) = \frac{x}{11}
\Rightarrow x = 11 \cdot \sin(60^\circ)
$$
$$
\sin(60^\circ) \approx 0.8660
\Rightarrow x \approx 11 \cdot 0.8660 = 9.526 \approx \boxed{9.5}
$$

✔ Answer: $x = 9.5$

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Problem 5

Given:
- Angle = 46°
- Adjacent side = 8
- Hypotenuse = $x$

Use cosine:
$$
\cos(46^\circ) = \frac{8}{x}
\Rightarrow x = \frac{8}{\cos(46^\circ)}
$$
$$
\cos(46^\circ) \approx 0.6947
\Rightarrow x \approx \frac{8}{0.6947} \approx 11.51 \approx \boxed{11.5}
$$

✔ Answer: $x = 11.5$

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Problem 6

Given:
- Angle = 43°
- Adjacent side = 22
- Hypotenuse = $x$

Use cosine:
$$
\cos(43^\circ) = \frac{22}{x}
\Rightarrow x = \frac{22}{\cos(43^\circ)}
$$
$$
\cos(43^\circ) \approx 0.7314
\Rightarrow x \approx \frac{22}{0.7314} \approx 29.94 \approx \boxed{29.9}
$$

✔ Answer: $x = 29.9$

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Problem 7

Given:
- Angle = 37°
- Hypotenuse = 10.3
- Opposite side = $x$

Use sine:
$$
\sin(37^\circ) = \frac{x}{10.3}
\Rightarrow x = 10.3 \cdot \sin(37^\circ)
$$
$$
\sin(37^\circ) \approx 0.6018
\Rightarrow x \approx 10.3 \cdot 0.6018 = 6.1985 \approx \boxed{6.2}
$$

✔ Answer: $x = 6.2$

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Problem 8

Given:
- Angle = 74°
- Opposite side = 19
- Adjacent side = $x$

Use tangent:
$$
\tan(74^\circ) = \frac{19}{x}
\Rightarrow x = \frac{19}{\tan(74^\circ)}
$$
$$
\tan(74^\circ) \approx 3.4874
\Rightarrow x \approx \frac{19}{3.4874} \approx 5.45 \approx \boxed{5.5}
$$

✔ Answer: $x = 5.5$

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✔ Final Answers (Rounded to Nearest Tenth):

1. $x = 6.8$
2. $x = 2.2$
3. $x = 6.0$
4. $x = 9.5$
5. $x = 11.5$
6. $x = 29.9$
7. $x = 6.2$
8. $x = 5.5$

Let me know if you'd like this formatted as a printable answer sheet!