Question

Suppose theta is an angle in the standard position whose terminal side is in Quadrant II and tan0= -sqrt3 Find the exact values of the remaining five trigonometric functions of theta

Answer 1

The exact values of the remaining five trigonometric functions of theta are

• sinθ = √3/2
• cosecθ = 2/√3
• cosθ = -1/2
• secθ = -2
• cotθ = -1/√3

Since tanθ = -√3.

The remaining five trigonometric functions of theta are sinθ, cosecθ, cosθ, secθ and cotθ.

The next trigonometric function of θ is cotθ.

cotθ = 1/tanθ

= 1/-√3

= -1/√3.

Also, tan²θ + 1 = sec²θ

Substituting tanθ = -√3 into the equation, we have

(-√3)² + 1 = sec²θ

3 + 1 = sec²θ

sec²θ = 4

secθ = ±√4

secθ = ±2

Since θ is in the quadrant II,

secθ = -2

Also, cosθ = 1/secθ

= 1/-2

= -1/2

Also, cot²θ + 1 = cosec²θ

Substituting cotθ = -1/√3 into the equation, we have

(-1/√3)² + 1 = cosec²θ

1/3 + 1 = cosec²θ

cosec²θ = 4/3

cosecθ = ±√(4/3)

cosecθ = ±2/√3

Since θ is in the quadrant II,

cosecθ = +2/√3

Also, sinθ = 1/cosecθ

= 1/2/√3

= √3/2

So, the exact values of the remaining five trigonometric functions of theta are

• sinθ = √3/2
• cosecθ = 2/√3
• cosθ = -1/2
• secθ = -2
• cotθ = -1/√3.

Learn more about trigonometric functions here:

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