Here are the step-by-step solutions for each problem on the worksheet. We will use trigonometry ratios (Sine, Cosine, and Tangent) to find the missing side $x$.
7. (turquoise blue)
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Identify sides: The angle is $37^\circ$. Side $x$ is
Opposite the angle. The side labeled 24 is the
Hypotenuse (longest side).
*
Choose Ratio: SOH (Sine = Opposite / Hypotenuse).
*
Equation: $\sin(37^\circ) = \frac{x}{24}$
*
Solve: Multiply both sides by 24.
$$x = 24 \cdot \sin(37^\circ)$$
$$x \approx 24 \cdot 0.6018$$
$$x \approx 14.44$$
8. (mint green)
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Identify sides: The angle is $47^\circ$. Side 36 is
Adjacent to the angle. Side $x$ is the
Hypotenuse.
*
Choose Ratio: CAH (Cosine = Adjacent / Hypotenuse).
*
Equation: $\cos(47^\circ) = \frac{36}{x}$
*
Solve: Multiply by $x$, then divide by $\cos(47^\circ)$.
$$x = \frac{36}{\cos(47^\circ)}$$
$$x \approx \frac{36}{0.682}$$
$$x \approx 52.79$$
9. (yellow)
*
Identify sides: The angle is $52^\circ$. Side 20 is
Adjacent to the angle. Side $x$ is the
Hypotenuse.
*
Choose Ratio: CAH (Cosine = Adjacent / Hypotenuse).
*
Equation: $\cos(52^\circ) = \frac{20}{x}$
*
Solve:
$$x = \frac{20}{\cos(52^\circ)}$$
$$x \approx \frac{20}{0.6157}$$
$$x \approx 32.48$$
10. (magenta)
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Identify sides: The angle is $18^\circ$. Side 26 is
Adjacent to the angle. Side $x$ is
Opposite the angle.
*
Choose Ratio: TOA (Tangent = Opposite / Adjacent).
*
Equation: $\tan(18^\circ) = \frac{x}{26}$
*
Solve: Multiply both sides by 26.
$$x = 26 \cdot \tan(18^\circ)$$
$$x \approx 26 \cdot 0.3249$$
$$x \approx 8.45$$
11. (lilac)
*
Identify sides: The angle is $21^\circ$. Side $x$ is
Adjacent to the angle. The side labeled 36 is the
Hypotenuse.
*
Choose Ratio: CAH (Cosine = Adjacent / Hypotenuse).
*
Equation: $\cos(21^\circ) = \frac{x}{36}$
*
Solve: Multiply both sides by 36.
$$x = 36 \cdot \cos(21^\circ)$$
$$x \approx 36 \cdot 0.9336$$
$$x \approx 33.61$$
12. (gray)
*
Identify sides: The angle is $45^\circ$. Side 25 is
Adjacent to the angle. Side $x$ is
Opposite the angle.
*
Choose Ratio: TOA (Tangent = Opposite / Adjacent).
*
Equation: $\tan(45^\circ) = \frac{x}{25}$
*
Solve: Multiply both sides by 25. Since $\tan(45^\circ) = 1$:
$$x = 25 \cdot 1$$
$$x = 25$$
Final Answer:
7. 14.44
8. 52.79
9. 32.48
10. 8.45
11. 33.61
12. 25
Parent Tip: Review the logic above to help your child master the concept of solving triangles worksheet.
