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Difficulty: MediumFundamental Trigonometric Identities

For an angle θ\theta in the third quadrant satisfying π<θ<3π2\pi < \theta < \frac{3\pi}{2}, the tangent value is tan⁡θ=43\tan \theta = \frac{4}{3}. What is the exact value of the expression sin⁡4θ−cos⁡4θ\sin^4 \theta - \cos^4 \theta?

Answer: 0.28

Answer

The exact numerical value of the expression is 0.28 (or 7/25).
By factoring sin⁡4θ−cos⁡4θ\sin^4 \theta - \cos^4 \theta as (sin⁡2θ−cos⁡2θ)(sin⁡2θ+cos⁡2θ)(\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta), we can apply the fundamental Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1. The expression simplifies cleanly to sin⁡2θ−cos⁡2θ\sin^2 \theta - \cos^2 \theta. Given tan⁡θ=43\tan \theta = \frac{4}{3} in Quadrant III, the reference triangle has opposite side 4, adjacent side 3, and hypotenuse 5. Thus, sin⁡θ=−45\sin \theta = -\frac{4}{5} and cos⁡θ=−35\cos \theta = -\frac{3}{5}. Substituting these values yields (−45)2−(−35)2=1625−925=725=0.28\left(-\frac{4}{5}\right)^2 - \left(-\frac{3}{5}\right)^2 = \frac{16}{25} - \frac{9}{25} = \frac{7}{25} = 0.28.

Step-by-Step Solution

1
Factor the fourth-degree trigonometric expression using difference of squares.
sin⁡4θ−cos⁡4θ=(sin⁡2θ−cos⁡2θ)(sin⁡2θ+cos⁡2θ)\sin^4 \theta - \cos^4 \theta = (\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta)
The difference of two squares a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b) applies directly to a=sin⁡2θa = \sin^2 \theta and b=cos⁡2θb = \cos^2 \theta.
2
Simplify using the fundamental Pythagorean trigonometric identity.
sin⁡4θ−cos⁡4θ=sin⁡2θ−cos⁡2θ\sin^4 \theta - \cos^4 \theta = \sin^2 \theta - \cos^2 \theta
By the Pythagorean identity, sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 for any angle θ\theta.
3
Determine sin⁡θ\sin \theta and cos⁡θ\cos \theta from the given quadrant and tangent value.
sin⁡θ=−45\sin \theta = -\frac{4}{5} and cos⁡θ=−35\cos \theta = -\frac{3}{5}
In Quadrant III (π<θ<3π2\pi < \theta < \frac{3\pi}{2}), both sine and cosine are negative. A standard 3-4-5 right triangle yields sin⁡θ=−45\sin \theta = -\frac{4}{5} and cos⁡θ=−35\cos \theta = -\frac{3}{5}.
4
Substitute the values into the simplified expression and compute the result.
(−45)2−(−35)2=1625−925=725=0.28\left(-\frac{4}{5}\right)^2 - \left(-\frac{3}{5}\right)^2 = \frac{16}{25} - \frac{9}{25} = \frac{7}{25} = 0.28
Squaring each trigonometric ratio yields positive values, giving a final simplified decimal value of 0.28.

Key Concept

Pythagorean Identity and Difference of Squares
Estimated Time:1m 30s
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Topics

Fundamental Trigonometric Identities

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