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Find sec theta if sin theta=(3sqrt13)/(13)

Find sec⁡θ \sec \theta if sin⁡θ=31313 \sin \theta=\frac{3 \sqrt{13}}{13}

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Full solution

Q. Find sec⁡θ \sec \theta if sin⁡θ=31313 \sin \theta=\frac{3 \sqrt{13}}{13}
  1. Understand Relationship: Understand the relationship between sec⁡(θ)\sec(\theta) and sin⁡(θ)\sin(\theta). Secant is the reciprocal of cosine, and sine and cosine are related through the Pythagorean identity: sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1. To find sec⁡(θ)\sec(\theta), we need to find cos⁡(θ)\cos(\theta) first and then take its reciprocal.
  2. Find Cosine: Use the given value of sin⁡(θ)\sin(\theta) to find cos⁡(θ)\cos(\theta). We are given sin⁡(θ)=31313\sin(\theta) = \frac{3\sqrt{13}}{13}. We can use the Pythagorean identity to find cos⁡(θ)\cos(\theta): cos⁡2(θ)=1−sin⁡2(θ)\cos^2(\theta) = 1 - \sin^2(\theta) cos⁡2(θ)=1−[31313]2\cos^2(\theta) = 1 - \left[\frac{3\sqrt{13}}{13}\right]^2
  3. Calculate Cosine^22: Calculate cos⁡2(θ)\cos^2(\theta).cos⁡2(θ)=1−[31313]2\cos^2(\theta) = 1 - \left[\frac{3\sqrt{13}}{13}\right]^2cos⁡2(θ)=1−[9⋅13132]\cos^2(\theta) = 1 - \left[\frac{9\cdot 13}{13^2}\right]cos⁡2(θ)=1−[913]\cos^2(\theta) = 1 - \left[\frac{9}{13}\right]cos⁡2(θ)=1313−913\cos^2(\theta) = \frac{13}{13} - \frac{9}{13}cos⁡2(θ)=413\cos^2(\theta) = \frac{4}{13}
  4. Find Cosine: Find cos⁡(θ)\cos(\theta).\newlineSince cos⁡2(θ)=413\cos^2(\theta) = \frac{4}{13}, we take the square root of both sides to find cos⁡(θ)\cos(\theta). However, we must consider both the positive and negative square roots because cosine can be positive or negative depending on the quadrant of θ\theta. Since we are not given information about the quadrant, we will assume θ\theta is in the first quadrant where cosine is positive:\newlinecos⁡(θ)=413\cos(\theta) = \sqrt{\frac{4}{13}}\newlinecos⁡(θ)=213\cos(\theta) = \frac{2}{\sqrt{13}}
  5. Rationalize Denominator: Rationalize the denominator.\newlineTo rationalize the denominator, we multiply the numerator and denominator by 13\sqrt{13}:\newlinecos⁡(θ)=213×1313\cos(\theta) = \frac{2}{\sqrt{13}} \times \frac{\sqrt{13}}{\sqrt{13}}\newlinecos⁡(θ)=21313\cos(\theta) = \frac{2\sqrt{13}}{13}
  6. Find Secant: Find sec⁡(θ)\sec(\theta).\newlineSecant is the reciprocal of cosine, so:\newlinesec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}\newlinesec⁡(θ)=1(21313)\sec(\theta) = \frac{1}{\left(\frac{2\sqrt{13}}{13}\right)}\newlinesec⁡(θ)=13213\sec(\theta) = \frac{13}{2\sqrt{13}}
  7. Rationalize Denominator: Rationalize the denominator of sec⁡(θ)\sec(\theta). To rationalize the denominator, we multiply the numerator and denominator by 13\sqrt{13}: sec⁡(θ)=13213×1313\sec(\theta) = \frac{13}{2\sqrt{13}} \times \frac{\sqrt{13}}{\sqrt{13}} sec⁡(θ)=13132×13\sec(\theta) = \frac{13\sqrt{13}}{2\times 13} sec⁡(θ)=132\sec(\theta) = \frac{\sqrt{13}}{2}

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