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WITHOUT using a calculator, determine the following in terms of sin 25°: 2.1.1 sin 335°, 2.1.2 cos 50°. Simplify the following expression to ONE trigonometric ratio: sin(-2x).(1-sin2 x)/sin(90°+x).tan x

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Answer

sin⁡335∘=−sin⁡25∘\sin 335^\circ = -\sin 25^\circ

cos⁡50∘=1−2sin⁡225∘\cos 50^\circ = 1 - 2\sin^2 25^\circ

−2cos⁡3x-2\cos^3 x

Step-by-step solution

12 steps

Here are the solutions:

2.1.1)

  1. Use the reduction formula for sine in the fourth quadrant, sin⁡(360∘−θ)=−sin⁡θ\sin(360^\circ - \theta) = -\sin \theta.

  2. Express 335∘335^\circ as 360∘−25∘360^\circ - 25^\circ.

    sin⁡335∘=sin⁡(360∘−25∘)\sin 335^\circ = \sin(360^\circ - 25^\circ)

  3. Apply the reduction formula.

    sin⁡335∘=−sin⁡25∘\sin 335^\circ = \boxed{-\sin 25^\circ}

    2.1.2)

  4. Use the double angle identity for cosine, cos⁡(2θ)=1−2sin⁡2θ\cos(2\theta) = 1 - 2\sin^2 \theta.

  5. Let 2θ=50∘2\theta = 50^\circ, so θ=25∘\theta = 25^\circ.

    cos⁡50∘=cos⁡(2×25∘)\cos 50^\circ = \cos(2 \times 25^\circ)

  6. Apply the double angle identity.

    cos⁡50∘=1−2sin⁡225∘\cos 50^\circ = \boxed{1 - 2\sin^2 25^\circ}

    2.2)

  7. Simplify each term in the expression using trigonometric identities.

    • sin⁡(−2x)=−sin⁡(2x)\sin(-2x) = -\sin(2x) (odd function identity)
    • 1−sin⁡2x=cos⁡2x1 - \sin^2 x = \cos^2 x (Pythagorean identity)
    • sin⁡(90∘+x)=cos⁡x\sin(90^\circ + x) = \cos x (co-function identity)
    • tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x} (quotient identity)

  8. Substitute these simplified terms into the expression.

    sin⁡(−2x)⋅(1−sin⁡2x)sin⁡(90∘+x)⋅tan⁡x=−sin⁡(2x)⋅cos⁡2xcos⁡x⋅sin⁡xcos⁡x\frac{\sin(-2x) \cdot (1 - \sin^2 x)}{\sin(90^\circ + x) \cdot \tan x} = \frac{-\sin(2x) \cdot \cos^2 x}{\cos x \cdot \frac{\sin x}{\cos x}}

  9. Simplify the denominator.

    −sin⁡(2x)⋅cos⁡2xsin⁡x\frac{-\sin(2x) \cdot \cos^2 x}{\sin x}

  10. Apply the double angle identity for sine, sin⁡(2x)=2sin⁡xcos⁡x\sin(2x) = 2\sin x \cos x.

    −(2sin⁡xcos⁡x)⋅cos⁡2xsin⁡x\frac{-(2\sin x \cos x) \cdot \cos^2 x}{\sin x}

  11. Cancel out sin⁡x\sin x from the numerator and denominator (assuming sin⁡x≠0\sin x \neq 0).

    −2cos⁡x⋅cos⁡2x-2\cos x \cdot \cos^2 x

  12. Combine the cosine terms.

    −2cos⁡3x\boxed{-2\cos^3 x}
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