Each figure shows an angle θ in standard position with it...

Question

Each figure shows an angle θ in standard position with its terminal side intersecting the unit circle. Evaluate the six circular function values of θ.

Unit circle with angle θ in standard position, showing terminal side intersecting at (√2/2, √2/2).

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the six circular function values of theta. The angle theta is in standard position and its terminal side intersects the unit circle at the specified point. Then we are given a graph. We have a corn plane, we have a vertical Y axis and a horizontal X axis. They come together at the origin in the middle. The range of our Y axis is from negative two to positive two. And the domain of our X axis is also from negative two to positive two. There is a unit circle where every point on the circle is exactly one unit away from the origin. And then we have our angle theta as told it's in standard position. The initial side is along the positive part of the X axis and the terminal side intersects the unit circle at the 0.9 divided by 41. That's the X value 40 divided by 41. That's our Y value were asked to determine the six circular function values of theta. Well, we recall from previous lessons that our six circular function values are sine cosign tangent because he can't, she can't and co tangents. So we want to find the sine of theta, the cosine of theta, the tangent of theta, the cosecant of theta, the S can of theta and the cotangent of theta. And we recall from previous lessons that when we're talking about angles that intersect the unit circle, the sine of that angle is going to be equal to the Y value where it intersects the cosine of that angle is going to be equal to the X value where that angle intersects the tangent of that angle is going to be equal to Y divided by X. The co can, the reciprocal function of sine is going to be equal to one divided by Y. The C can't, the reciprocal function of cosine is going to be equal to one divided by X and the cotangent, the reciprocal function of tangent is going to be equal to X divided by Y. So we know XX is equal to nine divided by 41. And we know yy is equal to 40 divided by 41. Now we just plug those in. So the sine of theta equals Y and Y equals 40 divided by 41. That's it. There's our sine of theta. The cosine of theta is equal to XX equals nine divided by 41. That's it. There's our cosine of theta. The tangent of theta equals Y divided by X. So Y is 40 divided by 41 divided by X which is nine divided by 41. When we're dividing fractions within fractions, we're multiplying by the reciprocal of the second fraction. So it's 40 divided by 41 multiplied by the reciprocal of nine divided by 41 which is 41 divided by nine. Now, these 40 ones cancel out and the tangent of theta equals 40 divided by nine. Now, because we're dealing with fractions here, all of the reciprocal fractions are just going to be the reciprocals uh or the reciprocal functions are going to be the reciprocals of these fractions. So for the KOC can't of theta, that's one divided by Y, we can flip our sin theta. So instead of 40 divided by 41 it's 41 divided by 40. And theres our KC can of the, for the C can of the, that's one divided by X or the reciprocal of our cosine. So instead of nine divided by 41 it's 41 divided by nine. And lastly, for our cotangent, that's X divided by Y or in other words, the reciprocal of our tangent function. So instead of 40 divided by nine, that's nine divided by 40 and there's our cotangent of the and that is it, there's our six circular function values of theta. Well done. We'll catch you on the next one.

Each figure shows an angle θ in standard position with its terminal side intersecting the unit circle. Evaluate the six circular function values of θ.

Key Concepts

Unit Circle and Standard Position

Circular Functions (Trigonometric Functions)

Evaluating Trigonometric Functions from Coordinates

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