Question

Find The 6 Trigonometric Functions Given The Following:

(Rationalize the denominators)
[ β=-√(5)/4 ; Q III.; ; θ=√(7)/3 ; Q IV. ]

Answer 1

The six trigonometric functions of the given angles β and θ are calculated by substituting the values into the trigonometric formulas. The values for sine, cosine, tangent, cosecant, secant, and cotangent are determined for each angle in their respective quadrants.

Step-by-step explanation:

To find the 6 trigonometric functions for the given angles β and θ, we need to substitute the values into the trigonometric formulas.

For β = -√5/4 in Quadrant III, we have:

• sine: sin(β) = -√5/4
• cosine: cos(β) = -√(1 - sin²(β)) = -√(1 - 5/16) = -√11/4
• tangent: tan(β) = sin(β) / cos(β) = (-√5/4) / (-√11/4) = √5/√11 = (√55)/11
• cosecant: csc(β) = 1 / sin(β) = 1 / (-√5/4) = -4/√5 = -4√5/5
• secant: sec(β) = 1 / cos(β) = 1 / (-√11/4) = -4/√11 = -4√11/11
• cotangent: cot(β) = 1 / tan(β) = 1 / (√5/√11) = √11/√5 = √11/5

For θ = √7/3 in Quadrant IV, we have:

• sine: sin(θ) = √7/3
• cosine: cos(θ) = -√(1 - sin²(θ)) = -√(1 - 7/9) = -√2/3
• tangent: tan(θ) = sin(θ) / cos(θ) = (√7/3) / (-√2/3) = -√7/√2 = -(√14/2)
• cosecant: csc(θ) = 1 / sin(θ) = 1 / (√7/3) = 3/√7 = (√7/7)
• secant: sec(θ) = 1 / cos(θ) = 1 / (-√2/3) = -3/√2 = -(√2/2)
• cotangent: cot(θ) = 1 / tan(θ) = 1 / (-(√14/2)) = -2/√14 = -(√14/7)