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: \[ \boxed{40^\circ} \]

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.

Consider the following triangle (not drawn to scale).Select the option that is the value of the angle marked θ.(Hint: you should not need to calculate anything as the given triangle is one of the standard triangles used to calculate useful known values of sine, cosine and tangent.)

Given up-pointing triangle△ABC, where A = 68degrees°, a = 6, and b = 9, find the measurement of angle B.

The diagram shows a triangle ABC with AC = 6 cm, BC = 8 cm, angle BAC = 60° and angle ABC = 𝛾. Find the exact value of sin⁡𝛾, simplifying your answer.