Question

36/sin^2 43°+4+sin^2 133°

Answer 1

$\frac{36}{sin^{2}43 + 4 + sin^{2}133} = \frac{36}{sin^{2}(30 + 13) + sin^{2}(120 + 13) + 4} = \frac{36}{\frac{(\sqrt{3}}{2}sin13 + \frac{1}{2}cos13)^2 + (\frac{(\sqrt{3}}{2}cos13 - \frac{1}{2}sin13)^2 + 4} = \frac{36}{sin^{2}13 + cos^{2}13 + 4} = \frac{36}{5}$

Answer 2

To simplify the given expression, we'll start by using the trigonometric identity:

sin^2(θ) + cos^2(θ) = 1

Since sin^2(θ) = 1 - cos^2(θ), we can rewrite the expression as follows:

36 / (1 - cos^2(43°)) + 4 + (1 - cos^2(133°))

Now, let's find the values of cos(43°) and cos(133°). We can use a calculator for this:

cos(43°) ≈ 0.7314 cos(133°) ≈ -0.6428

Now, substitute these values back into the expression:

36 / (1 - 0.7314^2) + 4 + (1 - (-0.6428)^2)

Calculate the values inside the parentheses first:

1 - 0.7314^2 ≈ 0.4684 1 - (-0.6428)^2 ≈ 0.5860

Now, the expression becomes:

36 / 0.4684 + 4 + 0.5860

Now, calculate each part of the expression:

36 / 0.4684 ≈ 76.8086 4 + 0.5860 ≈ 4.5860

Finally, add the calculated parts together:

76.8086 + 4.5860 ≈ 81.3946

So, the simplified value of the expression is approximately 81.3946.