Here are the step-by-step solutions for each problem on the worksheet.
1.
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Identify sides: We need to find $x$ (adjacent to the angle) and we have 88 (opposite). The hypotenuse is 89.
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Choose ratio: Tangent uses Opposite and Adjacent ($\tan = \frac{\text{Opp}}{\text{Adj}}$).
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Equation: $\tan(x^\circ) = \frac{88}{89}$
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Solve: $x = \tan^{-1}(\frac{88}{89}) \approx 44.67$
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Round: 45
2.
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Identify sides: We have the hypotenuse (93) and the adjacent side (82). We need the angle $w$.
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Choose ratio: Cosine uses Adjacent and Hypotenuse ($\cos = \frac{\text{Adj}}{\text{Hyp}}$).
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Equation: $\cos(w^\circ) = \frac{82}{93}$
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Solve: $w = \cos^{-1}(\frac{82}{93}) \approx 28.36$
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Round: 28
3.
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Identify sides: We have the hypotenuse (52) and the adjacent side (20). We need the angle $z$.
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Choose ratio: Cosine uses Adjacent and Hypotenuse.
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Equation: $\cos(z^\circ) = \frac{20}{52}$
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Solve: $z = \cos^{-1}(\frac{20}{52}) \approx 67.38$
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Round: 67
4.
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Identify sides: We have the angle $67^\circ$. We know the opposite side (26) and need the adjacent side ($y$).
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Choose ratio: Tangent uses Opposite and Adjacent.
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Equation: $\tan(67^\circ) = \frac{26}{y}$
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Rearrange: Multiply by $y$ and divide by $\tan(67^\circ)$ to get $y = \frac{26}{\tan(67^\circ)}$.
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Solve: $y \approx \frac{26}{2.356} \approx 11.03$
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Round: 11
5.
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Identify sides: We have the angle $34^\circ$. We know the adjacent side (40) and need the hypotenuse ($n$).
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Choose ratio: Cosine uses Adjacent and Hypotenuse.
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Equation: $\cos(34^\circ) = \frac{40}{n}$
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Rearrange: $n = \frac{40}{\cos(34^\circ)}$
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Solve: $n \approx \frac{40}{0.829} \approx 48.25$
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Round: 48
6.
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Identify sides: We have the opposite side (9) and the adjacent side (16). We need the angle $m$.
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Choose ratio: Tangent uses Opposite and Adjacent.
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Equation: $\tan(m^\circ) = \frac{9}{16}$
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Solve: $m = \tan^{-1}(\frac{9}{16}) \approx 29.36$
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Round: 29
Final Answer:
1. x = 45
2. w = 28
3. z = 67
4. y = 11
5. n = 48
6. m = 29
Parent Tip: Review the logic above to help your child master the concept of trigonometric ratios worksheet pdf.
