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Math  /  Trigonometry

QuestionSolve the equation. Write the numbers using integers or simplified fractions. sec⁡(−2x)=−2\sec (-2 x)=-\sqrt{2}
Part: 0/20 / 2
Part 1 of 2 (a) Write the solution set for the general solution. Use njn_{j} where nn is an integer.
The solution set for the general solution is {x∣x=\{x \mid x= □\square x=x= □\square 1\}. □\square

Studdy Solution

STEP 1

1. The equation involves the secant function, which is the reciprocal of the cosine function.
2. We will need to find the angle(s) that satisfy the given secant value.
3. The secant function is periodic, so there will be a general solution involving integer multiples of the period.

STEP 2

1. Convert the secant equation to a cosine equation.
2. Solve for the angle −2x-2x.
3. Determine the general solution for −2x-2x.
4. Solve for xx in terms of the general solution.

STEP 3

Convert the secant equation to a cosine equation. Recall that sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}. Thus, the equation sec⁡(−2x)=−2\sec(-2x) = -\sqrt{2} becomes:
cos⁡(−2x)=−12 \cos(-2x) = -\frac{1}{\sqrt{2}}

STEP 4

Recognize that −12-\frac{1}{\sqrt{2}} is equivalent to −22-\frac{\sqrt{2}}{2}, which is a known cosine value. The angles where cos⁡(θ)=−22\cos(\theta) = -\frac{\sqrt{2}}{2} are:
θ=3π4+2kπorθ=5π4+2kπ \theta = \frac{3\pi}{4} + 2k\pi \quad \text{or} \quad \theta = \frac{5\pi}{4} + 2k\pi
where kk is any integer.

STEP 5

Substitute −2x-2x for θ\theta in the general solutions:
1. −2x=3π4+2kπ-2x = \frac{3\pi}{4} + 2k\pi
2. −2x=5π4+2kπ-2x = \frac{5\pi}{4} + 2k\pi

STEP 6

Solve each equation for xx:
1. For −2x=3π4+2kπ-2x = \frac{3\pi}{4} + 2k\pi:
x=−3π8−kπ x = -\frac{3\pi}{8} - k\pi
2. For −2x=5π4+2kπ-2x = \frac{5\pi}{4} + 2k\pi:
x=−5π8−kπ x = -\frac{5\pi}{8} - k\pi
Thus, the solution set for the general solution is:
{x∣x=−3π8−kπ, x=−5π8−kπ where k is an integer} \{ x \mid x = -\frac{3\pi}{8} - k\pi, \, x = -\frac{5\pi}{8} - k\pi \, \text{where} \, k \, \text{is an integer} \}

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