Category 80 Example 1 Question 1 - mental math
Since cos A is 5/13, 13 is the hypotenuse and 5 is side adjacent to angle A. Since 5 and 13 are the ratio of Pythagorean triple 5:12:13, third side opposite to angle A must be 12. Hence, sin A = opposite/hypotenuse = 12/13.
Category 80 Example 1 Question 2 - no calculation required
cosine of an angle equals the sine of its complement.
Category 80 Example 2 Question 1 - mental math
Since the two triangles are similar, sin F = sin C. It is given that sin A is 0.8= 8/10. Hence, 10 is the hypotenuse and 8 is the side opposite to angle A. Since 8 and 10 are the ratio of Pythagorean triple 6:8:10, third side adjacent to angle A but opposite to angle C must be 6. Hence sin C = opposite/hypothenuse = 6/10 = 3/5 = sin F.