Question

examples: find the variable lengths

Explanation:

Step1: Identify trig - ratio for first triangle

We have a right - triangle with hypotenuse 17 and an angle of 26°. The side we want to find is opposite the given angle. So we use the sine ratio $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. So $\sin26^{\circ}=\frac{x}{17}$.

Step2: Solve for x in first triangle

Multiply both sides of the equation $\sin26^{\circ}=\frac{x}{17}$ by 17. We get $x = 17\times\sin26^{\circ}\approx17\times0.4384=7.45$.

Step3: Identify trig - ratio for second triangle

In the second triangle, we have an angle of 57°, the side adjacent to the angle is 20 and the hypotenuse is x. We use the cosine ratio $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. So $\cos57^{\circ}=\frac{20}{x}$.

Step4: Solve for x in second triangle

Cross - multiply to get $x\cos57^{\circ}=20$, then $x=\frac{20}{\cos57^{\circ}}\approx\frac{20}{0.5446}\approx36.72$.

Step5: Identify trig - ratio for third triangle

In the third triangle, with an angle of 52°, the side adjacent to the angle is 15 and the hypotenuse is x. Using the cosine ratio $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, we have $\cos52^{\circ}=\frac{15}{x}$.

Step6: Solve for x in third triangle

Cross - multiply: $x\cos52^{\circ}=15$, so $x=\frac{15}{\cos52^{\circ}}\approx\frac{15}{0.6157}\approx24.36$.

Step7: Identify trig - ratio for fourth triangle

In the fourth triangle, with an angle of 22°, the side opposite the angle is 12 and the hypotenuse is x. Using the sine ratio $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, we get $\sin22^{\circ}=\frac{12}{x}$.

Step8: Solve for x in fourth triangle

Cross - multiply: $x\sin22^{\circ}=12$, so $x=\frac{12}{\sin22^{\circ}}\approx\frac{12}{0.3746}\approx32.04$.

Step9: Identify trig - ratio for fifth triangle

In the fifth triangle, with an angle of 48°, the side adjacent to the angle is 14 and the side opposite is x. Using the tangent ratio $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, we have $\tan48^{\circ}=\frac{x}{14}$.

Step10: Solve for x in fifth triangle

Multiply both sides by 14: $x = 14\times\tan48^{\circ}\approx14\times1.1106 = 15.55$.

Step11: Identify trig - ratio for sixth triangle

In the sixth triangle, with an angle of 29°, the side opposite the angle is 18 and the hypotenuse is x. Using the sine ratio $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, we get $\sin29^{\circ}=\frac{18}{x}$.

Step12: Solve for x in sixth triangle

Cross - multiply: $x\sin29^{\circ}=18$, so $x=\frac{18}{\sin29^{\circ}}\approx\frac{18}{0.4848}\approx37.13$.

Answer:

For the first triangle: $x\approx7.45$; For the second triangle: $x\approx36.72$; For the third triangle: $x\approx24.36$; For the fourth triangle: $x\approx32.04$; For the fifth triangle: $x\approx15.55$; For the sixth triangle: $x\approx37.13$