Answer:
(-√10)/10
Step-by-step explanation:
The trig identity that relates the sine and the tangent is ...
sin(x) = ±tan(x)/√(1 + tan(x)²)
For your given value of tan(x), this evaluates to ...
sin(x) = ±(1/3)/(√(1 + (1/3)²) = ±1/√10 = (±√10)/10
The given value of the tangent function is positive, so the sine and cosine functions have the same sign (both are negative in this case).
So, your sine function is ...
sin(x) = (-√10)/10
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You can derive the above relationship using SOH CAH TOA.
Tan = Opposite/Adjacent
You are given the value of the tangent function as 1/3, so it is convenient to assign Opposite = 1 and Adjacent = 3. Then Hypotenuse is found using the Pythagorean theorem as ...
Hypotenuse = √(Opposite² + Adjacent²) = √(1² + 3²) = √10
Now, the sine is ...
Sin = Opposite/Hypotenuse = 1/√10
Rationalizing the denominator, this is ...
Sin = (√10)/10
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Now, you need to take into account the quadrant of the angle. Tangent is positive and Cosine is negative in the 3rd quadrant, where the Sine is also negative. Then your sine value is ...
sin(x) = (-√10)/10
