Here are the solutions to the exercises on the worksheet.
Exercise 1: Calculate the unknown side
To find the missing letter, we rearrange the formula so the letter is by itself. We use a calculator for the trigonometric values (sin, cos, tan).
a) $\sin 20 = \frac{a}{15}$
* Multiply both sides by 15: $a = 15 \times \sin 20$
* Calculation: $15 \times 0.3420... = 5.13$
*
Answer: $a \approx 5.13$
b) $\cos 37 = \frac{b}{30}$
* Multiply both sides by 30: $b = 30 \times \cos 37$
* Calculation: $30 \times 0.7986... = 23.96$
*
Answer: $b \approx 23.96$
c) $\tan 64 = \frac{c}{5}$
* Multiply both sides by 5: $c = 5 \times \tan 64$
* Calculation: $5 \times 2.0503... = 10.25$
*
Answer: $c \approx 10.25$
d) $\sin 52 = \frac{d}{32}$
* Multiply both sides by 32: $d = 32 \times \sin 52$
* Calculation: $32 \times 0.7880... = 25.22$
*
Answer: $d \approx 25.22$
e) $\cos 10 = \frac{e}{18}$
* Multiply both sides by 18: $e = 18 \times \cos 10$
* Calculation: $18 \times 0.9848... = 17.73$
*
Answer: $e \approx 17.73$
f) $\tan 25 = \frac{f}{22}$
* Multiply both sides by 22: $f = 22 \times \tan 25$
* Calculation: $22 \times 0.4663... = 10.26$
*
Answer: $f \approx 10.26$
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Exercise 2: Calculate the lengths of the lettered sides
We need to choose the right ratio: Sine ($\frac{\text{Opposite}}{\text{Hypotenuse}}$), Cosine ($\frac{\text{Adjacent}}{\text{Hypotenuse}}$), or Tangent ($\frac{\text{Opposite}}{\text{Adjacent}}$).
Triangle 1 (Top Left):
* Given: Angle $60^\circ$, Hypotenuse $12$. Find Adjacent side $a$.
* Use Cosine: $\cos 60 = \frac{a}{12}$
* $a = 12 \times \cos 60$
* $a = 12 \times 0.5 = 6$
*
Answer: $a = 6$ m
Triangle 2 (Top Middle):
* Given: Angle $45^\circ$, Adjacent side $8$. Find Hypotenuse $c$.
* Use Cosine: $\cos 45 = \frac{8}{c}$
* Rearrange: $c = \frac{8}{\cos 45}$
* $c = \frac{8}{0.7071...} = 11.31$
*
Answer: $c \approx 11.31$ m
Triangle 3 (Top Right):
* Given: Angle $25^\circ$, Hypotenuse $15$. Find Opposite side $e$.
* Use Sine: $\sin 25 = \frac{e}{15}$
* $e = 15 \times \sin 25$
* $e = 15 \times 0.4226... = 6.34$
*
Answer: $e \approx 6.34$ m
Triangle 4 (Bottom Left):
* Given: Angle $35^\circ$, Adjacent side $24$. Find Opposite side $b$.
* Use Tangent: $\tan 35 = \frac{b}{24}$
* $b = 24 \times \tan 35$
* $b = 24 \times 0.7002... = 16.80$
*
Answer: $b \approx 16.80$ m
Triangle 5 (Bottom Middle):
* Given: Angle $26^\circ$, Hypotenuse $15$. Find Adjacent side $d$.
* Use Cosine: $\cos 26 = \frac{d}{15}$
* $d = 15 \times \cos 26$
* $d = 15 \times 0.8988... = 13.48$
*
Answer: $d \approx 13.48$ m
Triangle 6 (Bottom Right):
* Given: Angle $70^\circ$, Adjacent side $17$. Find Hypotenuse $f$.
* Use Cosine: $\cos 70 = \frac{17}{f}$
* Rearrange: $f = \frac{17}{\cos 70}$
* $f = \frac{17}{0.3420...} = 49.71$
*
Answer: $f \approx 49.71$ m
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Exercise 3: Word Problem
*
Identify the parts: The wire is the hypotenuse ($30$ m). The angle with the ground is $25^\circ$. We want to find the height of the flagpole, which is the side
opposite the angle.
*
Choose Ratio: SOH CAH TOA $\rightarrow$ Sine ($\frac{\text{Opposite}}{\text{Hypotenuse}}$).
*
Equation: $\sin 25 = \frac{\text{height}}{30}$
*
Solve: $\text{height} = 30 \times \sin 25$
*
Calculation: $30 \times 0.4226... = 12.68$
Final Answer:
Exercise 1:
a) 5.13
b) 23.96
c) 10.25
d) 25.22
e) 17.73
f) 10.26
Exercise 2:
a = 6 m
b ≈ 16.80 m
c ≈ 11.31 m
d ≈ 13.48 m
e ≈ 6.34 m
f ≈ 49.71 m
Exercise 3:
The flagpole is approximately
12.68 meters high.
Parent Tip: Review the logic above to help your child master the concept of trigonometry word problems worksheet with answers.
