Answer:
64/27 or 2 10/27
Explanation:
To solve the given expression involving trigonometric functions, we'll start by determining the values of sine (sin), cosine (cos), tangent (tan), secant (sec), and cotangent (cot) based on the given value of cosθ.
Given: cosθ = 3/5
Using the Pythagorean identity, we can find the value of sinθ:
sinθ = √(1 - cos²θ)
= √(1 - (3/5)²)
= √(1 - 9/25)
= √(16/25)
= 4/5
Now that we know the values of sinθ and cosθ, we can determine the values of the other trigonometric functions:
tanθ = sinθ/cosθ
= (4/5)/(3/5)
= 4/3
secθ = 1/cosθ
= 1/(3/5)
= 5/3
cotθ = 1/tanθ
= 1/(4/3)
= 3/4
Now let's substitute these values into the given expression:
5 tan (π + θ) + 4 cos (π + θ)/5 sec (2π + θ) - 4 cot (2π + θ)
Substituting the trigonometric function values, we get:
5(4/3) + 4(-3/5) / 5(5/3) - 4(4/3)
Simplifying each term:
= 20/3 - 12/5 / 25/3 - 16/3
To simplify further, we need to find a common denominator for the fractions involved:
= (20/3 - 12/5) / (25/3 - 16/3)
Finding a common denominator of 15:
= (100/15 - 36/15) / (75/15 - 48/15)
= (64/15) / (27/15)
Dividing fractions:
= 64/15 * 15/27
= 64/27
Therefore, the simplified value of the expression is 64/27.