Find the exact value of tan2(Beta) if sin(Beta) = 5/13 (Beta in Quadrant II)
To find the exact value of tan(2β), we can use the double-angle identity for tangent:
tan(2β) = (2tanβ) / (1 - tan²β)
Given that sin(β) = 5/13 and β is in Quadrant II, we can draw a right triangle representing this situation. Βecause sin(β) = opposite / hypotenuse, we can assign the following values to the triangle:
opposite side = 5
hypotenuse = 13
Using the Pythagorean theorem, we can find the adjacent side:
a² + b² = c²
a² + 5² = 13²
a² + 25 = 169
a² = 144
a = √144
a = 12
Now that we have the values of the opposite and adjacent sides, we can determine the value of tan(β):
tan(β) = opposite / adjacent
tan(β) = 5/12
Using the double-angle identity:
tan(2β) = (2tanβ) / (1 - tan²β)
Substituting the value of tan(β):
tan(2β) = (2 * (5/12)) / (1 - (5/12)²)
tan(2β) = (10/12) / (1 - 25/144)
tan(2β) = (10/12) / (1 - 25/144)
tan(2β) = (10/12) / (144/144 - 25/144)
tan(2β) = (10/12) / (119/144)
tan(2β) = (10/12) * (144/119)
tan(2β) = 1440/1428
tan(2β) = 120/119
Therefore, the exact value of tan(2β) is 120/119.
To find the exact value of tan(2β), we can use the double-angle identity for tangent.
The double-angle identity for tangent is:
tan(2β) = (2tan(β)) / (1 - tan^2(β))
Given that sin(β) = 5/13 and β is in Quadrant II, we can find the value of tan(β) using the Pythagorean identity for sine and cosine:
sin^2(β) + cos^2(β) = 1
Since β is in Quadrant II, sin(β) is positive and cos(β) is negative.
sin^2(β) = 5^2 / 13^2 = 25/169
cos^2(β) = 1 - sin^2(β) = 1 - 25/169 = 144/169
Taking the square root of both sides:
sin(β) = 5/13
cos(β) = -12/13
Now we can find tan(β):
tan(β) = sin(β) / cos(β) = (5/13) / (-12/13) = -5/12
Using this value in the double-angle identity for tangent:
tan(2β) = (2tan(β)) / (1 - tan^2(β))
= (2*(-5/12)) / (1 - (-5/12)^2)
= (-10/12) / (1 - 25/144)
= (-10/12) / (119/144)
Multiplying the numerator and denominator by the reciprocal 144/119:
tan(2β) = (-10/12) * (144/119)
= -120/119
Therefore, the exact value of tan(2β) is -120/119.