Question

Triangle ABC is an isosceles right triangle, right angled at B. Find the value of sin² A + cos² C.

Answer 1

1. Identify triangle properties

In an isosceles right triangle ABC right-angled at B:

The angle at vertex B is ∠B = 90°.

Since it is isosceles, the two legs are equal (AB = BC), which means the acute angles must be equal: ∠A = ∠C.

The sum of angles in a triangle is 180°. 

Therefore, ∠A + ∠C = 180° – 90° = 90°.

This implies ∠A = ∠C = 45°.

2. Substitute angle values

The expression to evaluate is sin² A + cos² C.

Substituting A = 45° and C = 45°:

sin²(45°) + cos²(45°)

3. Calculate trigonometric values

We know the standard values for 45°:

`sin(45^circ) = 1/sqrt(2)`

`cos(45^circ) = 1/sqrt(2)`

Now, square these values:

`(1/sqrt(2))^2 + (1/sqrt(2))^2`

`1/2 + 1/2 = 1`

The value of sin² A + cos² C for an isosceles right triangle right-angled at B is 1.