Evaluate using the Pythagorean identities.
Find sinθ and cosθ if tanθ = 1/4 and sinθ >0
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Arora gave you a detailed solution to this type of question here:
https://www.jiskha.com/display.cgi?id=1516371278
follow the same steps, and stick to one name
well I suppose but I prefer the triangle
tan T = 1/4
but we know
tan^2 T + 1 = sec^2 T
so
1/16 + 1 = 1/cos^2 T
17/16 = 1/cos^2 T
so
cos^2 T = 16/17
take the square root now
then for the sin
sin^2 T = 1 - 16/17 = 1/17
take the square root now
===========================
check
T = tan^-1 .25
T = 14.04 deg
then
sin T = .243
sin^2 T = .0588
and 1/17 = .0588 so it worked
To evaluate sinθ and cosθ using the Pythagorean identities, we can follow these steps:
Step 1: Start with the given information: tanθ = 1/4 and sinθ > 0.
Step 2: Recall that tanθ is the ratio of sinθ to cosθ. Therefore, we can write tanθ = sinθ / cosθ.
Step 3: Substitute the given value of tanθ into the equation: 1/4 = sinθ / cosθ.
Step 4: Rearrange the equation to solve for sinθ: sinθ = (1/4) * cosθ.
Step 5: Recall the Pythagorean identity sin²θ + cos²θ = 1.
Step 6: Substitute the expression for sinθ obtained in step 4 into the Pythagorean identity: [(1/4) * cosθ]² + cos²θ = 1.
Step 7: Simplify the equation: (1/16) * cos²θ + cos²θ = 1.
Step 8: Combine like terms: (1/16 + 1) * cos²θ = 1.
Step 9: Simplify further: (17/16) * cos²θ = 1.
Step 10: Solve for cos²θ by dividing both sides of the equation by (17/16): cos²θ = 16/17.
Step 11: Take the square root of both sides of the equation to find cosθ: cosθ = ± √(16/17).
Since sinθ > 0 is given, we know that sine is positive. Therefore, to satisfy the given condition, we discard the negative value for cosθ.
Hence, sinθ = (1/4) and cosθ = √(16/17).