Question

❶ Find the values of x, y and z in each of the case given below ⟼

Answer 1

Answer:

1)

y = 45° (converse of alternate interior angle theorem)

z = 45° (alternate interior angle theorem)

k = 135° (exterior angle theorem)

2)

Cos θ + Sin θ = √2 Cos θ

(Cos θ + Sin θ)² = (√2 Cos θ)²

(Exponential property of equality)

Cos² θ + Sin² θ + 2 (Cos θ • Sin θ) = (√2 Cos θ)²

(Distributive property)

1 + 2(Cos θ • Sin θ) = (√2 Cos θ)²

(Definition of the pythagorean identity).

This identity is true because, if a² + b² = c². The triangle must be a right triangle, and a right triangle always has a 90° angle tangent to the triangle, and the tangent of 90° is 1, cos applied to the adjacent over hypotenuse, and sin applies to the opposite over hypotenuse, so this identity must work).

1 + 2(Cos θ • Sin θ) = 2 Cos² θ

(definition of an exponent)

1 + 2(Cos θ • Sin θ) – 1 = 2 Cos² θ – 1

(subtraction property of equality)

2(Cos θ • Sin θ) = 2 Cos² θ – 1

2(Sin θ • Cos θ) = 2 Cos² θ – 1

(Associative property of multiplication ; the input for both functions are θ so this rule works, this is an exception)

Sin (2x) = 2 Cos² θ – 1

(double angled identity of 2 sine and cos)