Given that $\sin x=\frac{1}{\sqrt{26}}$ and $\sin y=\frac{1}{\sqrt{5}}$ , and that angles $x$ and $y$ are both in Quadrant I, find the exact value of $\sin (x+y)$ , in simplest radical form. $\newline$ Answer:

View step-by-step help

Home

Math Problems

Algebra 2

Find trigonometric ratios using multiple identities

Full solution

Q. Given that

\sin x=\frac{1}{\sqrt{26}}

and

\sin y=\frac{1}{\sqrt{5}}

, and that angles

x

and

y

are both in Quadrant I, find the exact value of

\sin (x+y)

, in simplest radical form.

\newline

Answer:

Use Sine Addition Formula: To find $\sin(x+y)$ , we will use the sine addition formula: $\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)$ .
Find $\cos(x)$ and $\cos(y)$ : First, we need to find $\cos(x)$ and $\cos(y)$ . Since $x$ and $y$ are in Quadrant I, both $\cos(x)$ and $\cos(y)$ will be positive. We can use the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ to find $\cos(\theta)$ for both angles.
Calculate $\cos(x)$ : For angle $x$ , we have $\sin(x) = \frac{1}{\sqrt{26}}$ . So, $\sin^2(x) = \left(\frac{1}{\sqrt{26}}\right)^2 = \frac{1}{26}$ . Using the Pythagorean identity, we get $\cos^2(x) = 1 - \sin^2(x) = 1 - \frac{1}{26} = \frac{25}{26}$ . Therefore, $\cos(x) = \sqrt{\frac{25}{26}} = \frac{5}{\sqrt{26}}$ .
Calculate $\cos(y)$ : For angle $y$ , we have $\sin(y) = \frac{1}{\sqrt{5}}$ . So, $\sin^2(y) = \left(\frac{1}{\sqrt{5}}\right)^2 = \frac{1}{5}$ . Using the Pythagorean identity, we get $\cos^2(y) = 1 - \sin^2(y) = 1 - \frac{1}{5} = \frac{4}{5}$ . Therefore, $\cos(y) = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}}$ .
Substitute values into formula: Now we can substitute the values into the sine addition formula: $\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y) = \left(\frac{1}{\sqrt{26}}\right)\left(\frac{2}{\sqrt{5}}\right) + \left(\frac{5}{\sqrt{26}}\right)\left(\frac{1}{\sqrt{5}}\right)$ .
Simplify expression: Simplify the expression: $\sin(x+y) = \frac{2}{\sqrt{130}} + \frac{5}{\sqrt{130}} = \frac{2+5}{\sqrt{130}} = \frac{7}{\sqrt{130}}$ .
Rationalize the denominator: To rationalize the denominator, we multiply the numerator and the denominator by $\sqrt{130}$ : $\sin(x+y) = \frac{7}{\sqrt{130}} \times \frac{\sqrt{130}}{\sqrt{130}} = \frac{7\sqrt{130}}{130}$ .
Final simplification: Finally, we simplify the expression: $\sin(x+y) = \frac{7\sqrt{130}}{130}$ .