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Math  /  Trigonometry

QuestionSolve the equation 8tan⁡(4θ)+3=118 \tan (4 \theta)+3=11 for a value of θ\theta in the first quadrant. Give your answer in radians and degrees. θ=\theta= □\square radians, to 4 decimal places or □\square degrees.

Studdy Solution

STEP 1

1. The equation 8tan⁡(4θ)+3=11 8 \tan(4\theta) + 3 = 11 is trigonometric.
2. We are solving for θ\theta in the first quadrant, where 0<θ<π20 < \theta < \frac{\pi}{2}.
3. We will convert the solution from radians to degrees.

STEP 2

1. Isolate the trigonometric function tan⁡(4θ)\tan(4\theta).
2. Solve for 4θ4\theta.
3. Solve for θ\theta.
4. Convert θ\theta from radians to degrees.

STEP 3

Subtract 3 from both sides of the equation to isolate the term with the tangent function:
8tan⁡(4θ)+3=11 8 \tan(4\theta) + 3 = 11 8tan⁡(4θ)=11−3 8 \tan(4\theta) = 11 - 3 8tan⁡(4θ)=8 8 \tan(4\theta) = 8

STEP 4

Divide both sides by 8 to solve for tan⁡(4θ)\tan(4\theta):
tan⁡(4θ)=88 \tan(4\theta) = \frac{8}{8} tan⁡(4θ)=1 \tan(4\theta) = 1

STEP 5

Find 4θ4\theta by taking the arctangent of both sides. Since tan⁡(π4)=1\tan(\frac{\pi}{4}) = 1, we have:
4θ=π4 4\theta = \frac{\pi}{4}
Solve for θ\theta by dividing both sides by 4:
θ=π4×14 \theta = \frac{\pi}{4} \times \frac{1}{4} θ=π16 \theta = \frac{\pi}{16}

STEP 6

Convert θ\theta from radians to degrees. We know that 180∘=π180^\circ = \pi radians, so:
θ=π16×180∘π \theta = \frac{\pi}{16} \times \frac{180^\circ}{\pi} θ=180∘16 \theta = \frac{180^\circ}{16} θ=11.25∘ \theta = 11.25^\circ
The value of θ\theta is:
θ=π16 radians, or 11.25∘ \theta = \frac{\pi}{16} \text{ radians, or } 11.25^\circ

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