Question

Given that sin a/2 = 3/5 Given that sin a/2 is in quadrant II, determine the exact values of sin α,cos α,tan α,csc α,sec α, and cot α.

Answer 1

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.

1. 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.
2. 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.
3. 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).
4. Cosec α is the reciprocal of sin α, so csc α = 1 / sin α = 1 / (-3/5) = -5/3.
5. Sec α is the reciprocal of cos α, so sec α = 1 / cos α = 1 / (±4/5) = ±5/4.
6. 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

Answer 2

The exact trigonometric values for α using the given sine value in quadrant II are sin α = -24/25, cos α = 7/25, tan α = -24/7, csc α = -25/24, sec α = 25/7, and cot α = -7/24, calculated using the double angle formulas.

Step-by-step explanation:

Given that sin a/2 = 3/5 and it is in quadrant II, we can find the exact values of the trigonometric functions of α by first determining the cosine of α/2 and then using the double-angle formulas. In quadrant II, the cosine is negative. By applying the Pythagorean identity, we have cos a/2 = -√(1 - sin^2(a/2)), which gives us cos a/2 = -√(1 - (3/5)^2) = -√(1 - 9/25) = -√(16/25) = -4/5.

Using double-angle formulas, we get:

• sin α = 2sin(a/2)cos(a/2) = 2 * (3/5) * (-4/5) = -24/25
• cos α = cos^2(a/2) - sin^2(a/2) = (-4/5)^2 - (3/5)^2 = 16/25 - 9/25 = 7/25
• tan α = sin α/cos α = (-24/25) / (7/25) = -24/7
• csc α = 1/sin α = -25/24
• sec α = 1/cos α = 25/7
• cot α = cos α/sin α = (7/25) / (-24/25) = -7/24

These are the exact trigonometric values for α in quadrant II with the given sine value.