The angle theta_(1) is located in Quadrant I, and sin(theta_(1))=(17)/(20). What is the value of cos(theta_(1)) ? Express your answer exactly. cos(theta_(1))=

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The angle  theta_(1) is located in Quadrant  I, and  sin(theta_(1))=(17)/(20). What is the value of  cos(theta_(1)) ? Express your answer exactly.  cos(theta_(1))=

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Understand Relationship: Understand the relationship between sine and cosine in a right triangle. In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse, while the cosine is the ratio of the length of the adjacent side to the hypotenuse. For an angle θ₁ in the first quadrant, both sine and cosine values are positive. We can use the Pythagorean theorem to find the cosine if we know the sine.Use Pythagorean Identity: Use the Pythagorean identity to find cos(θ₁). The Pythagorean identity states that sin²(θ₁) + cos²(θ₁) = 1. We can solve for cos²(θ₁) by substituting the given value of sin(θ₁). sin²(θ₁) + cos²(θ₁) = 1 (17/20)² + cos²(θ₁) = 1Calculate sin²: Calculate sin²(θ₁). (17/20)² = (17²) / (20²) = 289 / 400Substitute and Solve: Substitute sin²(θ₁) into the Pythagorean identity and solve for cos²(θ₁). 289 / 400 + cos²(θ₁) = 1 cos²(θ₁) = 1 - 289 / 400 cos²(θ₁) = 400 / 400 - 289 / 400 cos²(θ₁) = (400 - 289) / 400 cos²(θ₁) = 111 / 400Find cos(θ₁): Find the value of cos(θ₁). Since θ₁ is in the first quadrant, cos(θ₁) will be positive. Therefore, we take the positive square root of cos²(θ₁). cos(θ₁) = √(111 / 400) cos(θ₁) = √111 / √400 cos(θ₁) = √111 / 20

$\newline$ The angle $\theta_{1}$ is located in Quadrant $\mathrm{I}$ , and $\sin \left(\theta_{1}\right)=\frac{17}{20}$ . $\newline$ What is the value of $\cos \left(\theta_{1}\right)$ ? $\newline$ Express your answer exactly. $\newline$ $\cos \left(\theta_{1}\right)=$

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