Trigonometic Ratios (SOH CAH TOA): Missing Side Lengths Worksheet ... - Free Printable

To solve these trigonometry problems, we use the SOH CAH TOA rules. These rules help us find missing sides in right-angled triangles.

* Sin = Opposite / Hypotenuse
* Cos = Adjacent / Hypotenuse
* Tan = Opposite / Adjacent

We need to identify which sides we know and which side we are trying to find (x) relative to the given angle.

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Section A

1)
* Angle: $35^{\circ}$
* Opposite side: $x$ (this is opposite the angle)
* Hypotenuse: $12 \text{ cm}$ (this is the longest side, opposite the right angle)
* Rule: SOH (Sine)
* Equation: $\sin(35^{\circ}) = \frac{x}{12}$
* Solve: $x = 12 \times \sin(35^{\circ})$
* Calculation: $x \approx 12 \times 0.57357 = 6.8829$
* Rounding: $6.88$

2)
* Angle: $67^{\circ}$
* Adjacent side: $x$ (this is next to the angle)
* Hypotenuse: $9 \text{ cm}$
* Rule: CAH (Cosine)
* Equation: $\cos(67^{\circ}) = \frac{x}{9}$
* Solve: $x = 9 \times \cos(67^{\circ})$
* Calculation: $x \approx 9 \times 0.39073 = 3.5166$
* Rounding: $3.52$

3)
* Angle: $49^{\circ}$
* Adjacent side: $16 \text{ cm}$
* Opposite side: $x$
* Rule: TOA (Tangent)
* Equation: $\tan(49^{\circ}) = \frac{x}{16}$
* Solve: $x = 16 \times \tan(49^{\circ})$
* Calculation: $x \approx 16 \times 1.15036 = 18.405$
* Rounding: $18.4$

4)
* Angle: $62^{\circ}$
* Hypotenuse: $17 \text{ cm}$
* Adjacent side: $x$
* Rule: CAH (Cosine)
* Equation: $\cos(62^{\circ}) = \frac{x}{17}$
* Solve: $x = 17 \times \cos(62^{\circ})$
* Calculation: $x \approx 17 \times 0.46947 = 7.981$
* Rounding: $7.98$

5)
* Angle: $27^{\circ}$
* Adjacent side: $5 \text{ cm}$
* Opposite side: $x$
* Rule: TOA (Tangent)
* Equation: $\tan(27^{\circ}) = \frac{x}{5}$
* Solve: $x = 5 \times \tan(27^{\circ})$
* Calculation: $x \approx 5 \times 0.50952 = 2.5476$
* Rounding: $2.55$

6)
* Angle: $33^{\circ}$
* Hypotenuse: $21 \text{ cm}$
* Opposite side: $x$
* Rule: SOH (Sine)
* Equation: $\sin(33^{\circ}) = \frac{x}{21}$
* Solve: $x = 21 \times \sin(33^{\circ})$
* Calculation: $x \approx 21 \times 0.54463 = 11.437$
* Rounding: $11.4$

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Section B

1)
* Angle: $15^{\circ}$
* Adjacent side: $14 \text{ cm}$
* Hypotenuse: $x$
* Rule: CAH (Cosine)
* Equation: $\cos(15^{\circ}) = \frac{14}{x}$
* Solve: $x = \frac{14}{\cos(15^{\circ})}$
* Calculation: $x \approx \frac{14}{0.9659} = 14.494$
* Rounding: $14.5$

2)
* Angle: $71^{\circ}$
* Opposite side: $19 \text{ cm}$
* Hypotenuse: $x$
* Rule: SOH (Sine)
* Equation: $\sin(71^{\circ}) = \frac{19}{x}$
* Solve: $x = \frac{19}{\sin(71^{\circ})}$
* Calculation: $x \approx \frac{19}{0.9455} = 20.094$
* Rounding: $20.1$

3)
* Angle: $42^{\circ}$
* Opposite side: $5 \text{ cm}$
* Adjacent side: $x$
* Rule: TOA (Tangent)
* Equation: $\tan(42^{\circ}) = \frac{5}{x}$
* Solve: $x = \frac{5}{\tan(42^{\circ})}$
* Calculation: $x \approx \frac{5}{0.9004} = 5.553$
* Rounding: $5.55$

4)
* Angle: $58^{\circ}$
* Adjacent side: $11 \text{ cm}$
* Hypotenuse: $x$
* Rule: CAH (Cosine)
* Equation: $\cos(58^{\circ}) = \frac{11}{x}$
* Solve: $x = \frac{11}{\cos(58^{\circ})}$
* Calculation: $x \approx \frac{11}{0.5299} = 20.758$
* Rounding: $20.8$

5)
* Angle: $20^{\circ}$
* Opposite side: $7 \text{ cm}$
* Adjacent side: $x$
* Rule: TOA (Tangent)
* Equation: $\tan(20^{\circ}) = \frac{7}{x}$
* Solve: $x = \frac{7}{\tan(20^{\circ})}$
* Calculation: $x \approx \frac{7}{0.36397} = 19.232$
* Rounding: $19.2$

6)
* Angle: $36^{\circ}$
* Adjacent side: $24 \text{ cm}$
* Hypotenuse: $x$
* Rule: CAH (Cosine)
* Equation: $\cos(36^{\circ}) = \frac{24}{x}$
* Solve: $x = \frac{24}{\cos(36^{\circ})}$
* Calculation: $x \approx \frac{24}{0.8090} = 29.666$
* Rounding: $29.7$

Final Answer:
Section A
1) 6.88
2) 3.52
3) 18.4
4) 7.98
5) 2.55
6) 11.4

Section B
1) 14.5
2) 20.1
3) 5.55
4) 20.8
5) 19.2
6) 29.7