An isosceles triangle has a height of 8 and base of 30. What are all three angles of this triangle?

Given an isosceles triangle with a height of 8 and a base of 30, we can use trigonometric ratios to find the angles.

Since the triangle is isosceles, the base angles are congruent. Let's denote one of these base angles as ( \theta ).

Using trigonometric ratios in a right triangle, we can find ( \theta ) by considering the height and half of the base, which forms a right triangle with the base angle.

[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{\frac{30}{2}} = \frac{8}{15} ]

Using the inverse sine function, we can find ( \theta ):

[ \theta = \arcsin\left(\frac{8}{15}\right) ]

This yields one of the base angles.

Since the triangle is isosceles, the sum of the angles in a triangle is ( 180^\circ ), and since we have two equal base angles, we can find the third angle by subtracting twice the base angle from ( 180^\circ ):

[ \text{Third angle} = 180^\circ - 2 \times \theta ]

This provides the measure of all three angles in the isosceles triangle.