How do you find the values of the six trigonometric functions when given that cos(θ) = 4/5 and θ lies in the third quadrant?

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To find the values of the six trigonometric functions given that cos(θ) = 4/5 and that the angle θ lies in the third quadrant, we can follow these steps:

Step 1: Understanding the Signs of Trigonometric Functions in the Third Quadrant

In the third quadrant (from 180° to 270°), the cosine and sine functions are negative, while the tangent function is positive.

Step 2: Finding the Value of sin(θ)

We can use the Pythagorean identity, which states:

sin²(θ) + cos²(θ) = 1

Substituting the given value of cos(θ):

sin²(θ) + (4/5)² = 1

This simplifies to:

sin²(θ) + 16/25 = 1

Next, we can subtract 16/25 from both sides:

sin²(θ) = 1 - 16/25

Converting 1 to a fraction with 25 as the denominator gives us:

sin²(θ) = 25/25 - 16/25 = 9/25

Taking the square root of both sides:

sin(θ) = ±√(9/25) = ±3/5

Since we are in the third quadrant, where sin is negative, we have:

sin(θ) = -3/5

Step 3: Finding the Value of tan(θ)

The tangent function is the ratio of the sine to the cosine:

tan(θ) = sin(θ)/cos(θ)

Substituting the values we have:

tan(θ) = (-3/5) / (4/5) = -3/4

However, since both sine and cosine are negative in the third quadrant, tan(θ) is positive:

tan(θ) = 3/4

Step 4: Finding the Values of the Other Three Trigonometric Functions

The reciprocal functions are as follows:

csc(θ) = 1/sin(θ) = 1/(-3/5) = -5/3
sec(θ) = 1/cos(θ) = 1/(4/5) = 5/4
cot(θ) = 1/tan(θ) = 1/(3/4) = 4/3

Summary of the Six Trigonometric Functions

  • sin(θ) = -3/5
  • cos(θ) = 4/5
  • tan(θ) = 3/4
  • csc(θ) = -5/3
  • sec(θ) = 5/4
  • cot(θ) = 4/3

In conclusion, we have successfully determined the values of the six trigonometric functions for the angle θ given the constraints.

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