If a=6 in. and θ=45° , find the value of c. Round your answer to the nearest hundredth.

Answer 1

To find the value of $c$ given that $a=6$ inches and $\theta ={45}^{\circ }$, we can use trigonometric relationships. However, the nature of $c$ must be specified; generally, it could refer to the length of a side in a right triangle where $a$ is one of the legs, and $\theta$ is the angle opposite that leg.

For example, if we are referring to a right triangle, we would use the sine, cosine, or tangent functions. Assuming $c$ is the hypotenuse, we can use the sine function:

$$\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{c}$$

Substituting the known values:

$$\sin ({45}^{\circ })=\frac{6}{c}$$

Since $\sin ({45}^{\circ })=\frac{\sqrt{2}}{2}$:

$$\frac{\sqrt{2}}{2}=\frac{6}{c}$$

Now, rearranging for $c$:

$$c=\frac{6}{\frac{\sqrt{2}}{2}}=6\cdot \frac{2}{\sqrt{2}}=\frac{12}{\sqrt{2}}$$

To rationalize the denominator:

$$c=\frac{12\sqrt{2}}{2}=6\sqrt{2}$$

Now we need to evaluate $6\sqrt{2}$:

$$\sqrt{2}\approx 1.414$$ $$c\approx 6\cdot 1.414\approx 8.484$$

Rounding to the nearest hundredth, we get:

$$c\approx 8.48\text{ in.}$$

Therefore, the value of $c$ is $\boxed{{\displaystyle 8.48}}$ inches.