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.

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine the six circular function values of theta, the angle theta is in standard position and the terminal side intersects the unit circle at the specified point. We're given a coordinate plane, we have a vertical Y axis and a horizontal X axis. They come together at the origin in the middle. The domain of our X axis is from negative two to positive two. And the range of our Y axis is from negative two to positive two as well. We have a unit circle on our coordinate plane. Every point of the unit circle is exactly one unit away from the origin. And then we have an angle theta in standard position. Its initial side is along the positive part of the X axis and then it heads counterclockwise and intersects the unit circle in quadrant three at the point negative 12 divided by 37. That's our X value negative 35 divided by 37. That's our Y value. So determining the six circular functions of data, we recall from previous lessons that the six circular functions are the sign of the, the cosine of theta, the tangent of theta, the Kosi can't of theta, the sea can of theta in the cotangent of the. And we recall from previous lessons that when we're talking about an angle that intersects with the unit circle, the sign of theta is equal to Y, the cosine of theta is equal to X. The tangent of theta is equal to Y divided by X. The cosecant of theta. That's the reciprocal function of the sine function is one divided by Y. The C of theta, which is the reciprocal function of the cosine function is one divided by X and the co tangent of theta. That's the reciprocal function of our tangent of theta is X divided by Y. Well, we know XX equals negative 12 divided by 37. And we know, yy equals negative 35 divided by 37. So we just need to plug in for our XS and Ys and we're good to go. So the sine of theta equals Y which is negative 35 divided by 37. And that's it. We've solved for the sine of theta. The cosine of theta equals X which is negative 12 divided by 37. Thats our coign of theta. The tangent of theta is Y divided by XY is negative 35 divided by 37 divided by X which is negative 12 divided by 37. When we're dividing fractions, that's the same as multiplying by the reciprocal of the second fraction So this is negative 35 divided by 37 multiplied by negative 37 divided by 12. Our 30 sevens cancel out and our negatives cancel out multiplying a negative by a negative. That's a positive. So our tangent of theta equals 35 divided by 12. Now, for our reciprocal functions, we can just take the reciprocals of their reciprocal functions fractions. So for the Kosi can of theta, that's one divided by Y or the reciprocal of the sine function. So the reciprocal of negative 35 divided by 37 is negative 37 divided by 35. There's our koi can of the, for the C can of theta, that's the reciprocal function of our cosine cosine of theta is negative 12 divided by 37. So the reciprocal of that is negative 37 divided by 12. And there's our C can of theta. And finally, cotangent is the reciprocal of tangent. Tangent is 35 divided by 12. So cotangent is 12 divided by 35. And that's it. That'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.

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Key Concepts

Unit Circle

Trigonometric Functions

Standard Position of an Angle

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