I want the explanation in English

Here's how the trigonometric values for 30^, 45^, and 60^ are derived using special right-angled triangles. Step 1: Deriving values for 45^ We use an isosceles right-angled triangle. Consider a right-angled triangle with two equal sides, each of length 1. The angles will be 45^, 45^, and 90^. Using the Pythagorean Theorem, the length of the hypotenuse (h) is: h = sqrt(1^2 + 1^2) = sqrt(1 + 1) = sqrt(2) Now we can calculate the trigonometric values for 45^: 45^ = oppositehypotenuse = (1)/(sqrt(2)) 45^ = adjacenthypotenuse = (1)/(sqrt(2)) 45^ = oppositeadjacent = (1)/(1) = 1 Step 2: Deriving values for 30^ and 60^ We use an equilateral triangle with side lengths of 2. All angles are 60^. Draw an altitude from one vertex to the midpoint of the opposite side. This altitude bisects the top angle and the base, creating two 30^-60^-90^ right-angled triangles. In each right-angled triangle: • The hypotenuse is 2. • The side opposite the 30^ angle (or adjacent to the 60^ angle) is 1 (half of the equilateral triangle's side). • Using the Pythagorean Theorem, the length of the side opposite the 60^ angle (or adjacent to the 30^ angle) is h: h^2 + 1^2 = 2^2 h^2 + 1 = 4 h^2 = 3 h = sqrt(3) Step 3: Calculating values for 30^ and 60^ For the 30^ angle: • Opposite side is 1. • Adjacent side is sqrt(3). • Hypotenuse is 2. 30^ = oppositehypotenuse = (1)/(2) 30^ = adjacenthypotenuse = sqrt(3)2 30^ = oppositeadjacent = (1)/(sqrt(3)) For the 60^ angle: • Opposite side is sqrt(3). • Adjacent side is 1. • Hypotenuse is 2. 60^ = oppositehypotenuse = sqrt(3)2 60^ = adjacenthypotenuse = (1)/(2) 60^ = oppositeadjacent = sqrt(3)1 = sqrt(3) These derivations show how the standard trigonometric values for these special angles are obtained using basic geometry and the Pythagorean Theorem. That's 2 down. 3 left today — send the next one.