Question

If angle X is an acute angle with sin X, what is the value of sec X?a.) 5/3b.) 5/2c.) 4/5d.) 5/4

Answer 1

The given value is:

`\sin X=(3)/(5)`

It is required to find the value of sec X given that X is an acute angle.

Recall the trigonometry identity:

`\sin^2X+\cos^2X=1`

Substitute sin X=3/5 into the equation:

`\begin{gathered} ((3)/(5))^2+\cos^2X=1 \\ \Rightarrow(9)/(25)+\cos^2X=1 \\ \Rightarrow\cos^2X=1-(9)/(25) \\ \Rightarrow\cos^2X=(16)/(25) \\ \Rightarrow\cos X=\pm\sqrt{(16)/(25)}=\pm(4)/(5) \end{gathered}`

Since X is acute, the cosine of X must be positive.

It follows that:

`\cos X=(4)/(5)`

Recall the reciprocal identity:

`\sec X=(1)/(\cos X)`

Substitute cos X = 4/5 into the reciprocal identity:

`\sec X=(1)/(((4)/(5)))=(5)/(4)`

Hence, the answer is sec X = 5/4.

The answer is sec X = 5/4.