In a right triangle with acute angle θ, the side adjacent to θ has length 3 and the hypotenuse has length 9. Find the length of the side opposite of θ.

Given a right triangle, solve for θ.

To find the size of the angle \(\theta\) in the right-angled triangle, we can use trigonometric ratios. Given that we have the length of the adjacent side (6) and the hypotenuse (8), we can use the cosine function: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{6}{8} = \frac{3}{4} \] Now, we need to find the angle \(\theta\) whose cosine is \(\frac{3}{4}\). We can use the inverse cosine function (also known as arccos): \[ \theta = \cos^{-1}\left(\frac{3}{4}\right) \] Using a calculator to find the inverse cosine: \[ \theta \approx \cos^{-1}(0.75) \approx 41.4^\circ\] Therefore, the size of the angle \(\theta\) to one decimal place is: \[ \boxed{41.4^\circ} \]

To find the size of the angle \(\theta\) in the triangle with sides of lengths 3, 6, and 8, we can use the Law of Cosines. The Law of Cosines states: \[ c^2 = a^2 + b^2 - 2ab \cos(\theta) \] In this case, \(a = 3\), \(b = 8\), and \(c = 6\). We need to find the angle \(\theta\) opposite the side of length 6. 1. Substitute the given values into the Law of Cosines: \[ 6^2 = 3^2 + 8^2 - 2 \cdot 3 \cdot 8 \cdot \cos(\theta) \] 2. Simplify the equation: \[ 36 = 9 + 64 - 48 \cos(\theta) \] 3. Combine like terms: \[ 36 = 73 - 48 \cos(\theta) \] 4. Isolate \(\cos(\theta)\): \[ 36 - 73 = -48 \cos(\theta) \] \[ -37 = -48 \cos(\theta) \] \[ \cos(\theta) = \frac{37}{48} \] 5. Use the inverse cosine function to find \(\theta\): \[ \theta = \cos^{-1}\left(\frac{37}{48}\right) \] Using a calculator to find the inverse cosine: \[ \theta \approx \cos^{-1}\left(\frac{37}{48}\right) \approx 40^\circ \] Therefore, the size of the angle \(\theta\) to the nearest degree is: \[ \boxed{40^\circ} \]

If sec⁡ θ = √2 and cot⁡ θ < 0, then find the exact value of cot ⁡θ.*−(√2)√2−11In ∆ABC, ∠A is a right angle and m∠B=45. How long is side BC if AB+AC=16?*8√2 units4√2 units16√2 units2√2 unitsIn a right triangle with acute angle θ, the side adjacent to θ has length 3 and the hypotenuse has length 9. Find the value of csc θ.*(3/4)√2(2√2)/331/3

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