Monday, June 25, 2018

trigonometry - Find $sin(A)$ and $cos(A)$ given $cos^4(A) - sin^4(A) = frac{1}{2}$ and $A$ is located in the second quadrant.




Question: Find $\sin(A)$ and $\cos(A)$, given $\cos^4(A)-\sin^4(A)=\frac{1}{2}$ and $A$ is located in the second quadrant.




Using the fundamental trigonometric identity, I was able to find that:



• $\cos^2(A) - \sin^2(A) = \frac{1}{2}$




and



• $$ \cos(A) \cdot \sin(A) = -\frac{1}{4} $$



However, I am unsure about how to find $\sin(A)$ and $\cos(A)$ individually after this.



Edit: I solved the problem through using the Fundamental Trignometric Identity with the difference of $\cos^2(A)$ and $\sin^2(A)$.


Answer



Hint




$$\left( \cos(A)+ \sin(A) \right)^2 = 1+2 \sin(A) \cos(A)=\frac{1}{2} \\
\left( \cos(A)- \sin(A) \right)^2 = 1-2 \sin(A) \cos(A)=\frac{3}{2} $$



Take the square roots, and pay attention to the quadrant and the fact that $\cos^4(A) >\sin^4(A)$ to decide is the terms are positive or negative.



Alternate simpler solution
$$2 \cos^2(A)= \left( \cos^2(A)+\sin^2(A)\right)+\left( \cos^2(A)-\sin^2(A)\right)=1+\frac{1}{2} \\
2 \sin^2(A)= \left( \cos^2(A)+\sin^2(A)\right)-\left( \cos^2(A)-\sin^2(A)\right)=1-\frac{1}{2} \\$$


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