2sinsqaredpi/6+cosecsqared7pi/6cossqaredpi/3=3/2 in trignometry
To solve the given trigonometric equation, we need to simplify the expression on the left-hand side (LHS) and then manipulate it to match the right-hand side (RHS).
Let's simplify the LHS step-by-step:
1. Start by converting the given angles from degrees to radians:
pi/6 = 30 degrees
7pi/6 = 210 degrees
pi/3 = 60 degrees
2. Write the equation with the converted angles:
2sin^2(30) + csc^2(210)cos^2(60) = 3/2
3. Evaluate the trigonometric functions at the given angles:
sin(30) = 1/2
cos(60) = 1/2
csc(210) = -sqrt(3)/2 (reciprocal of sin(210))
4. Substitute the values in the equation:
2(1/2)^2 + (-sqrt(3)/2)^2 * (1/2)^2 = 3/2
5. Simplify the expression further:
1/2 + 3/4 * 1/4 = 3/2
6. Continue simplifying:
1/2 + 3/16 = 3/2
7. Find a common denominator and combine the fractions:
8/16 + 3/16 = 3/2
8. Add the fractions:
11/16 = 3/2
9. Multiply both sides of the equation by 16 to eliminate the fraction:
11 = 24
Since 11 is not equal to 24, the equation is not true.
Therefore, the given equation 2sin^2(pi/6) + csc^2(7pi/6)cos^2(pi/3) = 3/2 is not satisfied in trigonometry.