if tan theta=7/24 and sin theta is greater than 0 what is the exact value of sin theta
draw a triangle. If tanθ > 0 and sinθ > 0 then we are in the first quadrant.
the two legs are 7 and 24, the hypotenuse is 25
sinθ = 7/25
To find the exact value of sin(theta) given that tan(theta) = 7/24 and sin(theta) is greater than 0, we can use the trigonometric identity:
tan(theta) = sin(theta) / cos(theta)
In this case, we know that tan(theta) = 7/24. We can rewrite this equation as:
7/24 = sin(theta) / cos(theta)
Since sin(theta) is greater than 0, we know that the sine function is positive in the given quadrant. To determine the sign of cos(theta), we can use the Pythagorean identity:
sin^2(theta) + cos^2(theta) = 1
Since sin(theta) is positive, sin^2(theta) is also positive. Therefore, cos^2(theta) must be less than 1 for the equation to hold true. This means that cos(theta) is also positive.
Now, we can solve for sin(theta) by rearranging the equation:
7/24 = sin(theta) / (sqrt(1 - sin^2(theta)))
To simplify further, we can square both sides of the equation:
(7/24)^2 = [sin(theta) / (sqrt(1 - sin^2(theta))))^2
49/576 = sin^2(theta) / (1 - sin^2(theta))
Next, we can rearrange the equation and multiply both sides by (1 - sin^2(theta)):
49/576 * (1 - sin^2(theta)) = sin^2(theta)
Expand the left side of the equation:
49/576 - 49/576 * sin^2(theta) = sin^2(theta)
Add 49/576 * sin^2(theta) to both sides of the equation:
49/576 = (1 + 49/576) * sin^2(theta)
Multiply both sides of the equation by 576:
49 = 625 * sin^2(theta)
Divide both sides of the equation by 625:
sin^2(theta) = 49/625
Take the square root of both sides of the equation:
sin(theta) = sqrt(49/625)
Simplify the right side of the equation:
sin(theta) = 7/25
Therefore, the exact value of sin(theta) is 7/25.