In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. W...

Question

In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c.

sin(-t - 2𝜋) - cos(-t - 4𝜋) - tan(-t - 𝜋)

Accepted Answer

Hello everybody. I I, today, today we're going to look again at this question that states rewrite the following expression in terms of H K and M, if sign of A is equal to H cosine of A is equal to K and tangents of A is equal to M where the expression given is negative sign of open parentheses, negative A minus four pi closed parentheses plus cosine of open parentheses, negative A minus eight pi closed parentheses minus tangent of open parentheses, negative A minus pi closed parentheses. Now the answers provided are A 4h plus eight K plus M B negative 4h plus eight K minus MC H plus K plus M N D negative H plus K minus M no. Before we start rearranging this and rewriting our expression, let's make a note of something important here. So let's note that sign of theta. So sign of theta plus two pi sign of open parentheses, theta plus two pi closed parentheses is equal to sign of theta. And I'll explain what I'm noting in a second cosign of the open parentheses. Theta plus two pi close parentheses is sequel to cosine theta and tangent of open parentheses, theta plus pi closed parentheses is equal to tangent beta. Now, why is this? Well, this has to do with the periods of each trigonometric expression. Now, let's what does that mean? So we have sign sign of theta cosine of theta, the and S data have a period of two pie. Now, what does this mean? It means that every two pi units, the expression of sign cosine cos or second will repeat Now for tangent data and cotangent data, those have a period of pie. So it's the same thing. Every pi units, the tangent expression or the code 10 expression would be that is why if you have theta plus pi that's the same thing of saying tangent data because it's just the start of the next period. So that's why we also write sign and cosine of zero plus two pi it's the same as saying sign of of data or cosine of data because it's just the start of the next period. Now write sign cosine and tenure because that's those are the ones we're dealing with. So let's rewrite our expression. We have a negative sign of open parentheses, negative A minus four pi closed parentheses plus cosine of open parentheses, negative A minus eight pi closed parentheses minus tangents of open parentheses, negative A minus pi closed parentheses. Now this will be equal to, if we remove a negative out of the first expression, we'll have negative open bracket, negative sign of open parentheses. A plus four pi closed parentheses, closed bracket plus cosine of open parentheses. A plus eight pie, noun, closed parentheses, minus open bracket. Negative tangent of open parentheses. A plus P. Now closed parentheses, closed bracket. So all we did here is factor out some negatives so we can get this into a different form or the form that we're more accustomed to of having a theta plus a period. Now, this will be equal to sine of A and this is because we have a plus four pi, which is the same for that sign. We have that theta plus two pi is sign of beta four pi is just the same thing as saying two periods. So it's just the start of another period. So it's the same thing as saying a sign of A and we have that plus. Now we have cosine of a plus eight pi, we saw that a period of cosine is two pi. So if we have a pi the, that's four periods. So it's still the start of a brand new period. So we can rewrite it as cosine of A. And finally, we have plus and we have tangents of A plus pi and we noted that a period of tangent is pi so this can be rewritten as plus tangents of A. So using what they told us at the beginning where they told us that sign of A is H cosine of A S K and tangent of A S M, we can rewrite this as H plus K plus M or answer choice C in this case. So I hope that was helpful. And until next time.

In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c.
sin(-t - 2𝜋) - cos(-t - 4𝜋) - tan(-t - 𝜋)

Key Concepts

Trigonometric Function Identities for Negative Angles

Periodicity of Trigonometric Functions

Expressing Trigonometric Functions in Terms of Given Variables

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