Question

determine the length of c to the nearest tenth of a centimetre. 3 cm 67° c 6.5 cm 7.6 cm 7.7 cm 7.8 cm

Explanation:

Step1: Identify trigonometric relation

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 67^{\circ}$, the opposite side to the angle $\theta$ is $3$ cm and the hypotenuse is $c$. So, $\sin67^{\circ}=\frac{3}{c}$.

Step2: Solve for $c$

We know that $\sin67^{\circ}\approx0.9205$. Then, $c=\frac{3}{\sin67^{\circ}}$. Substituting the value of $\sin67^{\circ}$, we get $c=\frac{3}{0.9205}\approx3.26$ (this is wrong approach. Let's use cosine). Using $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\cos67^{\circ}=\frac{3}{c}$. Since $\cos67^{\circ}\approx0.3907$, then $c = \frac{3}{\cos67^{\circ}}$.

Step3: Calculate the value of $c$

$c=\frac{3}{0.3907}\approx7.7$ cm.

Answer:

C. 7.7 cm