Final answer:
To find the values of sin α, cos α, tan α, csc α, sec α, and cot α, we can use the given information that sin(a/2) = 3/5 and a/2 is in Quadrant II. By applying trigonometric identities and using the properties of quadrants, we can determine the exact values: sin α = -3/5, cos α = ±4/5 (positive or negative depending on the sign of sin α), tan α = -3/4 or 3/4 (depending on the sign of cos α), csc α = -5/3, sec α = ±5/4, cot α = ±4/3.
Step-by-step explanation:
To find the values of sin α, cos α, tan α, csc α, sec α, and cot α, we need to use the given information that sin(a/2) = 3/5 and a/2 is in Quadrant II.
- Since sin(a/2) = 3/5 and a/2 is in Quadrant II, we know that sin α = -sin(a/2) = -(3/5) = -3/5.
- We can use the Pythagorean identity to find cos α. Since sin α = -3/5, we have cos α = ±√(1 - sin² α) = ±√(1 - (3/5)²) = ±√(1 - 9/25) = ±√(16/25) = ±4/5.
- Tan α is given by tan α = sin α / cos α = (-3/5) / (±4/5) = -3/4 or 3/4 (depending on the sign of cos α).
- Cosec α is the reciprocal of sin α, so csc α = 1 / sin α = 1 / (-3/5) = -5/3.
- Sec α is the reciprocal of cos α, so sec α = 1 / cos α = 1 / (±4/5) = ±5/4.
- Cot α is the reciprocal of tan α, so cot α = 1 / tan α = 1 / (±3/4) = ±4/3.
Therefore, the exact values of sin α, cos α, tan α, csc α, sec α, and cot α are:
- sin α = -3/5 (negative because it is in Quadrant II)
- cos α = ±4/5 (positive or negative depending on the sign of sin α)
- tan α = -3/4 or 3/4 (depending on the sign of cos α)
- csc α = -5/3
- sec α = ±5/4
- cot α = ±4/3