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Find all angles, 0^(@) <= theta < 360^(@), that satisfy the equation below, to the nearest tenth of a degree. 4sec^(2)theta-25=0 Answer: theta=

Find all angles, 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newline4sec⁡2θ−25=0 4 \sec ^{2} \theta-25=0 \newlineAnswer: θ= \theta=

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Csc, sec, and cot of special anglesright chevron icon

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Q. Find all angles, 0∘≤θ<360∘ 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newline4sec⁡2θ−25=0 4 \sec ^{2} \theta-25=0 \newlineAnswer: θ= \theta=
  1. Solve for sec⁡(θ)\sec(\theta): Solve the equation for sec⁡(θ)\sec(\theta).
    4sec⁡2(θ)−25=04\sec^2(\theta) - 25 = 0
    Add 2525 to both sides.
    4sec⁡2(θ)=254\sec^2(\theta) = 25
    Divide both sides by 44.
    sec⁡2(θ)=254\sec^2(\theta) = \frac{25}{4}
    Take the square root of both sides.
    sec⁡(θ)=±254\sec(\theta) = \pm\sqrt{\frac{25}{4}}
    sec⁡(θ)=±52\sec(\theta) = \pm\frac{5}{2}
  2. Find cosine values: Find the corresponding cosine values since sec⁡(θ)\sec(\theta) is the reciprocal of cos⁡(θ)\cos(\theta).sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}cos⁡(θ)=1sec⁡(θ)\cos(\theta) = \frac{1}{\sec(\theta)}cos⁡(θ)=±25\cos(\theta) = \pm\frac{2}{5}
  3. Determine cosine quadrants: Determine the quadrants where cosine is positive and negative.\newlineCosine is positive in the first and fourth quadrants and negative in the second and third quadrants.
  4. Principal angle for positive cosine: Use the inverse cosine function to find the principal angle for the positive cosine value. \newlineθ=cos⁡−1(25)\theta = \cos^{-1}(\frac{2}{5})\newlineCalculate the principal angle.\newlineθ≈cos⁡−1(0.4)\theta \approx \cos^{-1}(0.4)\newlineθ≈66.4∘\theta \approx 66.4^\circ
  5. Second angle for positive cosine: Find the second angle in the first cycle (0∘0^\circ to 360∘360^\circ) where cosine is positive.\newlineSince cosine is also positive in the fourth quadrant, the second angle is 360∘−66.4∘360^\circ - 66.4^\circ.\newlineθ≈360∘−66.4∘\theta \approx 360^\circ - 66.4^\circ\newlineθ≈293.6∘\theta \approx 293.6^\circ
  6. Principal angle for negative cosine: Use the inverse cosine function to find the principal angle for the negative cosine value.\newlineθ=cos⁡−1(−25)\theta = \cos^{-1}(-\frac{2}{5})\newlineCalculate the principal angle.\newlineθ≈cos⁡−1(−0.4)\theta \approx \cos^{-1}(-0.4)\newlineθ≈113.6∘\theta \approx 113.6^\circ
  7. Second angle for negative cosine: Find the second angle in the first cycle (0∘0^\circ to 360∘360^\circ) where cosine is negative.\newlineSince cosine is also negative in the third quadrant, the second angle is 180∘+(180∘−113.6∘)180^\circ + (180^\circ - 113.6^\circ).\newlineθ≈180∘+(180∘−113.6∘)\theta \approx 180^\circ + (180^\circ - 113.6^\circ)\newlineθ≈180∘+66.4∘\theta \approx 180^\circ + 66.4^\circ\newlineθ≈246.4∘\theta \approx 246.4^\circ

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