Here are the step-by-step solutions for each problem on the worksheet. We will use trigonometric ratios (Sine, Cosine, Tangent) to find the missing sides.
Reminder of Ratios (SOH CAH TOA):
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Sine = Opposite / Hypotenuse
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Cosine = Adjacent / Hypotenuse
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Tangent = Opposite / Adjacent
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1) Find $f$
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Identify sides: Angle is $56^\circ$. Side $f$ is
Opposite. The side labeled 40 is the
Hypotenuse (longest side).
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Choose ratio: SOH $\rightarrow$ Sine.
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Equation: $\sin(56^\circ) = \frac{f}{40}$
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Solve: $f = 40 \times \sin(56^\circ)$
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Calculation: $40 \times 0.8290 \approx 33.16$
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Round: $33.2$
2) Find $y$
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Identify sides: Angle is $81^\circ$. Side 24 is
Adjacent. Side $n$ is the
Hypotenuse. *Note: The variable in the image is labeled 'n', but the question asks for 'y'. We will solve for the hypotenuse 'n'.*
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Choose ratio: CAH $\rightarrow$ Cosine.
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Equation: $\cos(81^\circ) = \frac{24}{n}$
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Solve: $n = \frac{24}{\cos(81^\circ)}$
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Calculation: $\frac{24}{0.1564} \approx 153.45$
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Round: $153.5$
3) Find $o$
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Identify sides: Angle is $37^\circ$. Side $o$ is
Opposite. Side 12 is
Adjacent.
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Choose ratio: TOA $\rightarrow$ Tangent.
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Equation: $\tan(37^\circ) = \frac{o}{12}$
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Solve: $o = 12 \times \tan(37^\circ)$
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Calculation: $12 \times 0.7536 \approx 9.04$
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Round: $9.0$
4) Find $o$
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Identify sides: Angle is $35^\circ$. Side 24 is
Opposite. Side $o$ is
Adjacent.
*
Choose ratio: TOA $\rightarrow$ Tangent.
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Equation: $\tan(35^\circ) = \frac{24}{o}$
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Solve: $o = \frac{24}{\tan(35^\circ)}$
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Calculation: $\frac{24}{0.7002} \approx 34.27$
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Round: $34.3$
5) Find $j$
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Identify sides: Angle is $36^\circ$. Side $j$ is
Adjacent. Side 45 is the
Hypotenuse.
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Choose ratio: CAH $\rightarrow$ Cosine.
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Equation: $\cos(36^\circ) = \frac{j}{45}$
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Solve: $j = 45 \times \cos(36^\circ)$
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Calculation: $45 \times 0.8090 \approx 36.40$
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Round: $36.4$
6) Find $v$
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Identify sides: Angle is $34^\circ$. Side $v$ is
Opposite. Side 37 is the
Hypotenuse.
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Choose ratio: SOH $\rightarrow$ Sine.
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Equation: $\sin(34^\circ) = \frac{v}{37}$
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Solve: $v = 37 \times \sin(34^\circ)$
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Calculation: $37 \times 0.5592 \approx 20.69$
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Round: $20.7$
7) Find $v$
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Identify sides: Angle is $38^\circ$. Side $v$ is
Adjacent. Side 45 is the
Hypotenuse.
*
Choose ratio: CAH $\rightarrow$ Cosine.
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Equation: $\cos(38^\circ) = \frac{v}{45}$
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Solve: $v = 45 \times \cos(38^\circ)$
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Calculation: $45 \times 0.7880 \approx 35.46$
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Round: $35.5$
8) Find $f$
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Identify sides: Angle is $48^\circ$. Side 20 is
Opposite. Side $f$ is
Adjacent.
*
Choose ratio: TOA $\rightarrow$ Tangent.
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Equation: $\tan(48^\circ) = \frac{20}{f}$
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Solve: $f = \frac{20}{\tan(48^\circ)}$
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Calculation: $\frac{20}{1.1106} \approx 18.008$
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Round: $18.0$
Final Answer:
1) f = 33.2
2) y = 153.5
3) o = 9.0
4) o = 34.3
5) j = 36.4
6) v = 20.7
7) v = 35.5
8) f = 18.0
Parent Tip: Review the logic above to help your child master the concept of trigonometric ratios in right triangles worksheet.
