2 trig questions

sdcardz

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May 17, 2019
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1. Value of cos2theta when tantheta = 3/4 and pi < theta < 3pi/2
2. Solve 5cos^2x + 4 cos x = 1 if 0 <= x <= 2pi
 
1.) Suppose you have a right triangle, and the side opposite \(\theta\) is 3 units in length, while the side adjacent to \(\theta\) is 4 units in length. What is the length of the hypotenuse?
 
To follow up:

From the Pythagorean theorem, we find the hypotenuse \(h\) is:

[MATH]h=\sqrt{3^2+4^2}=5[/MATH]
Using a double-angle identity for cosine, we may write:

[MATH]\cos(2\theta)=1-2\sin^2(\theta)=1-2\left(\frac{-3}{5}\right)^2=\frac{7}{25}[/MATH]
For the 2nd problem, we can write as a quadratic in \(\cos(x)\) in standard form:

[MATH]5\cos^2(x)+4\cos(x)-1=0[/MATH]
Factor:

[MATH](5\cos(x)-1)(\cos(x)+1)=0[/MATH]
From the first factor, we find:

[MATH]x\in\left\{\arccos\left(\frac{1}{5}\right),2\pi-\arccos\left(\frac{1}{5}\right)\right\}[/MATH]
And from the second factor we find:

[MATH]x\in\{\pi\}[/MATH]
 
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