Question 1: Calculate the exact value of sin 30°.

Answer: 1/2

Question 2: Calculate cos 45°.

Answer: √2/2

Question 3: Find sin 60°.

Answer: √3/2

Question 4: Express cos 45° as a simplified fraction.

Answer: √2/2

Question 1: Calculate sin 60° and cos 30°, then verify that sin² θ + cos² θ = 1 for θ = 60°.

Answer: sin 60° = √3/2, cos 30° = √3/2, sum of squares = 1

Question 2: Given a right-angled triangle with angles 45° and 45°, if the hypotenuse is 10, find the length of each leg.

Answer: Legs = 10/√2 = 5√2

Question 3: Using exact values, find tan 45°. Show your working.

Answer: 1

Question 4: Construct a triangle with one angle 30°, another 60°, and the side opposite 30° of length 6 units. Calculate the length of the side opposite 60°.

Answer: Side = 6√3

Question 5: Explain why sin 45° equals cos 45° using the unit circle or right triangle reasoning.

Answer: In an isosceles right triangle, the two non-hypotenuse sides are equal, so sine and cosine for 45° are equal.

Question 1: A ladder leans against a wall forming a 60° angle with the ground. If the ladder is 8 meters long, how high does it reach on the wall?

Answer: Height = 8 * √3/2 = 4√3 meters

Question 2: A flagpole makes a 45° angle with the ground from a point 10 meters away from its base. Calculate the approximate height of the flagpole.

Answer: Height = 10 * tan 45° = 10 meters

Question 1: A triangle has angles 30°, 60°, and 90°. The side opposite 30° is 5 units. Find the length of the hypotenuse and the side opposite 60°, explaining your reasoning.

Answer: Hypotenuse = 10 units, side opposite 60° = 5√3 units

Question 2: Derive the exact value of sin 75° using angle addition formulas involving 45° and 30°, showing all steps.

Answer: sin 75° = sin (45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6/4) + (√2/4) = (√6 + √2)/4

Question 1: A student calculates cos 45° as 1/√2. Identify and correct the mistake in their calculation.

Answer: They should rationalize the denominator: 1/√2 = √2/2

Question 2: Without a calculator, find sin 30° + cos 60°, and verify if the result equals 1.

Answer: sin 30° + cos 60° = 1/2 + 1/2 = 1

Question 3: Explain why sin 45° and cos 45° are equal based on unit circle definitions.

Answer: On the unit circle, the coordinates at 45° are (√2/2, √2/2), so sine and cosine are equal for that angle.