If cos x = -0.750 and sin ≈ 0.6614, then tan x/2 ≈ ....

Question

If cos x = -0.750 and sin ≈ 0.6614, then tan x/2 ≈ .

Accepted Answer

Welcome back. I am so glad you're here. We're asked to find the approximate value of the tangent of theta divided by two. If the sine of theta equals 0.818 and the cosine of theta equals negative 0.575. Our answer choices are answer choice. A negative 1.925. Answer choice B 1.925. Answer choice C negative 0.326 and answer choice D 0.326. All right. So we have the tangent of theta divided by two which looks a lot like a half angle. So we can use our half angle identities to solve here. So we recall from previous lessons for our half angle identities that we have multiple ones for tangent. But I'm choosing one. It's OK. If you choose a different one, we have the tangent of a divided by two equals the one I chose is equal to one minus the cosine of A, all divided by the sine of A. And since we know that our A is going to be theta here, we know what our cosine of theta and our sine of theta are equal to, we just need to plug these things in. So the tangent of theta divided by two equals in the numerator, we have one minus the cosine of instead of a, it's theta which is negative 0.575. And then that's divided by the sine of theta which is 0.818. So in the numerator, one minus negative 0.575 that's going to be one plus 0.575 which is 1.575 divide that by 0.818 can plug that into a calculator. And you will get approximately and this is rounding 1.9254 and it keeps going. But all of our answer choices are rounded to three decimal places. So this five looks to the four remains a five and we'll get an answer of a positive 1.9 to 5. And that's the approximate value of the tangent of theta divided by two when sine equals that and cosine equals that. So we look at our answer choices and that matches with answer choice. B well done. We'll catch you on the next one.

If cos x = -0.750 and sin ≈ 0.6614, then tan x/2 ≈ .

Key Concepts

Relationship Between Sine, Cosine, and Tangent

Half-Angle Formulas

Sign Determination Based on Quadrants

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