Find the value of sin-1⁡(35)+sin-1⁡(45)+cos-1⁡(3√2)

The correct option is 2π/3.

To find the value of the given expression, let us first identify and clarify the standard mathematical representation of the terms in the expression:
1. The term sin−1(35) represents the inverse sine function sin−135.
2. The term sin−1(45) represents sin−145.
3. The term cos−1(3√2) represents cos−132.

Substituting these clarified terms, the expression becomes:
E=sin−135+sin−145+cos−132

Step 1: Simplify the first two terms
Let θ=sin−145. This means that:
sin(θ)=45

Using the fundamental trigonometric identity sin2(θ)+cos2(θ)=1, we can find cos(θ):
cos(θ)=1−sin2(θ)=1−452=1−1625=925=35

Therefore, we have:
θ=cos−135

This shows that:
sin−145=cos−135

Now, substitute this back into the sum of the first two terms:
sin−135+sin−145=sin−135+cos−135

Using the standard identity sin−1(x)+cos−1(x)=π2 (for |x|≤1), we get:
sin−135+cos−135=π2

Step 2: Evaluate the third term
Next, we evaluate:
cos−132

Since cosπ6=32, we have:
cos−132=π6

Step 3: Combine all terms
Adding the simplified values together:
E=π2+π6

Find a common denominator to sum the fractions:
E=3π6+π6=4π6=2π3

Thus, the final value of the expression is 2π3.