Question

Given sec A= sqrt(53)/2 and that angle A is in Quadrant IV, find the exact value of sin Ain simplest radical form using a rational denominator.

Answer 1

`\sin A=-(7√(53))/(53)`

Explanation:

Given value of sec A:

`\sec A=(√(53))/(2)`

To find the value of sin A, we can use the following trigonometric identities:

`\boxed{\begin{array}{c}\underline{\textsf{Trigonometric Identities }}\\\\\sec x=(1)/(\cos x)\\\\\sin^2x + \cos^2x=1\end{array}}`

Find cos A:

`\sec A=(√(53))/(2)\implies (1)/(\cos A)=(√(53))/(2)\implies \cos A=(2)/(√(53))`

Substitute cos A into the trigonometric identity and solve for sin A:

`\begin{aligned}\sin^2A+\left((2)/(√(53))\right)^2&=1\\\\\sin^2A+(4)/(53)&=1\\\\\sin^2A&=1-(4)/(53)\\\\\sin^2A&=(53-4)/(53)\\\\\sin^2A&=(49)/(53)\\\\\sin A&=\pm\sqrt{(49)/(53)}\\\\\sin A&=\pm (7)/(√(53))\\\\\sin A&=\pm (7√(53))/(53)\end{aligned}`

As sine is negative in Quadrant IV, then:

`\large\boxed{\boxed{\sin A=-(7√(53))/(53)}}`