(iv) Cos^(-1)(4)/(5)+sin^(-1)(3)/(5)=sin^(-1)(27)/(11)

Jasonbradley

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Jun 16, 2024

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Proving the trigonometric equation: cos^(-1)(4/5) + sin^(-1)(3/5) = sin^(-1)(27/25)

This article will prove the trigonometric equation: cos^(-1)(4/5) + sin^(-1)(3/5) = sin^(-1)(27/25). We will use the following trigonometric identities and concepts:

• sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
• cos(sin^(-1)(x)) = √(1 - x^2)
• sin(cos^(-1)(x)) = √(1 - x^2)

Step 1: Let's define variables.

Let α = cos^(-1)(4/5) and β = sin^(-1)(3/5).

Step 2: Using the identities to find sin(α) and cos(β).

From the definition of α, we know cos(α) = 4/5. Using the identity sin(cos^(-1)(x)) = √(1 - x^2), we can find sin(α) as follows:

sin(α) = sin(cos^(-1)(4/5)) = √(1 - (4/5)^2) = √(9/25) = 3/5

Similarly, from the definition of β, we know sin(β) = 3/5. Using the identity cos(sin^(-1)(x)) = √(1 - x^2), we can find cos(β) as follows:

cos(β) = cos(sin^(-1)(3/5)) = √(1 - (3/5)^2) = √(16/25) = 4/5

Step 3: Using the angle addition formula to find sin(α + β).

Now, using the angle addition formula, we can find sin(α + β):

sin(α + β) = sin(α)cos(β) + cos(α)sin(β) = (3/5)(4/5) + (4/5)(3/5) = 12/25 + 12/25 = 24/25

Step 4: Finding the inverse sine of sin(α + β).

Therefore, α + β = sin^(-1)(24/25).

Step 5: Finalizing the proof.

Since we initially defined α = cos^(-1)(4/5) and β = sin^(-1)(3/5), we can substitute them back into the equation:

cos^(-1)(4/5) + sin^(-1)(3/5) = sin^(-1)(24/25)

Conclusion:

We have successfully proven that cos^(-1)(4/5) + sin^(-1)(3/5) = sin^(-1)(24/25) using trigonometric identities and the angle addition formula. This demonstrates the relationship between different trigonometric functions and their inverses.