Given $\tan (\theta)=-\frac{3}{4}$ where $\frac{\pi}{2}<\theta<\pi$, find the exact value of $\cos (2 \theta)$.

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alessandra5169A alessandra5169A

03-11-2026
Mathematics

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Given $\tan (\theta)=-\frac{3}{4}$ where $\frac{\pi}{2}<\theta<\pi$, find the exact value of $\cos (2 \theta)$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T22:42:57-05:00

Analyze the given trigonometric condition: $\tan(\theta) = -\frac{3}{4}$ where $\theta$ is in the second quadrant.
Utilize the identity $\cos(2\theta) = \frac{1 - \tan^2(\theta)}{1 + \tan^2(\theta)}$ to directly solve for $\cos(2\theta)$.
Compute $\tan^2(\theta) = \frac{9}{16}$.
Substitute the value to find $\cos(2\theta) = \frac{1 - 9/16}{1 + 9/16} = \boxed{\frac{7}{25}}$.

Explanation

Analysis of the Problem

The problem asks for the exact value of $\cos(2\theta)$ given $\tan(\theta) = -\frac{3}{4}$ for $\frac{\pi}{2} < \theta < \pi$. In the second quadrant ($\frac{\pi}{2} < \theta < \pi$), tangent is negative, which is consistent with the given value of $-\frac{3}{4}$. To find $\cos(2\theta)$, we can use trigonometric identities that relate the double angle of cosine to the tangent of the angle.

Relevant Trigonometric Identities

A convenient identity relating the double angle of cosine to the tangent is:
\[\cos(2\theta) = \frac{1 - \tan^2(\theta)}{1 + \tan^2(\theta)}\]
This formula is derived from $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$ and the fact that $\cos^2(\theta) = \frac{1}{1+\tan^2(\theta)}$ and $\sin^2(\theta) = \frac{\tan^2(\theta)}{1+\tan^2(\theta)}$. We already have $\tan(\theta) = -\frac{3}{4}$, so we can directly substitute this into the identity.

Calculation of the Value

First, we calculate $\tan^2(\theta)$:
\[\tan^2(\theta) = \left(-\frac{3}{4}\right)^2 = \frac{9}{16}\]
Next, we substitute this into the formula for $\cos(2\theta)$:
\[\cos(2\theta) = \frac{1 - \frac{9}{16}}{1 + \frac{9}{16}}\]
To simplify the expression, we perform the arithmetic:
Numerator: $1 - \frac{9}{16} = \frac{16}{16} - \frac{9}{16} = \frac{7}{16}$
Denominator: $1 + \frac{9}{16} = \frac{16}{16} + \frac{9}{16} = \frac{25}{16}$
Thus,
\[\cos(2\theta) = \frac{\frac{7}{16}}{\frac{25}{16}} = \frac{7}{16} \cdot \frac{16}{25} = \frac{7}{25}\]

Conclusion

The exact value of $\cos(2\theta)$ is determined to be $\frac{7}{25}$. Given the constraints of the second quadrant, this value is consistent. The final answer is $\boxed{\frac{7}{25}}$.

Examples

In engineering, such as calculating the stress distribution on a beam at a certain angle relative to a force, you often encounter double-angle trigonometric identities. If the slope of a component is known (represented by the tangent of the angle), these identities allow you to find the projected forces or moments on doubled-angle axes without needing to solve for the specific angle $\theta$ itself. This is essential for analyzing rotating shafts or complex truss structures efficiently.