A point P(x, y) is shown on the unit circle corresponding to a...

Question

A point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.

Accepted Answer

Welcome back. Everyone. In this problem. A point P negative 11 divided by 61 and 60 divided by 61 is plotted on the unit circle for a certain angle. Theta determine the values of the trigonometric functions. Sine theta cast theta tangent, theta cotangent, theta S theta and cos theta corresponding to this point. Here we have a diagram of our unit circle. And for our answer choices, it says sign theater is negative 11 divided by 61 C. Theater is 60 divided by 61 than theater is negative 60 divided by 11. Coin theater equals negative 11 divided by 60 C. Can theta equals negative 61 divided by 11 and cos theta equals 61 divided by 60 B says it's 60 divided by 61 negative 11 divided by 61 negative 60 divided by 11 negative 11 divided by 6061 divided by 60 negative 61 divided by 11 respectively. C says it's 60 divided by 61 negative 11 divided by 61 negative 60 divided by 11 negative 11 divided by 60 negative 61 divided by 11 and 61 divided by 60. And the D says it's 60 divided by 61 negative 11 divided by 61 negative 11 divided by 60 negative 60 divided by 11, negative 61 divided by 11 and 61 divided by 60. Now here, if we're given a unit circle, what do we know about the unit circle that can help us? Well, recall, OK. That on the unit circle, on the unit circle, OK. The XY coordinates, OK are equal to the cosine of theta and the sine of theta. So if we can find the XY coordinates at that point, then we know what the cosine of theta and the sine of theta will be. And then we'll be able to use those to figure out the rest. So let's start with those values. Let's go to our point. OK. So here notice that the X coordinate is negative 11 divided by 61 and the Y coordinate is 60 divided by 61. So what does this tell us? But from our unit circle, then we can tell that sin of theta. OK is equal to 60 divided by 61. And the cosine of theta is equal to negative the X value which is negative 11 divided by 61. Now, what about the tangent of theta? Well, we aren't giving it, giving it explicitly on the unit circle but we know that the tangent is the ratio of the sine function to the cosine function. So if we substitute those values OK, then it's going to be equal to 60 divided by 61 divided by negative 11 divided by 61 or we can multiply by the reciprocal which is uh ne sorry multiplied by negative 61 divided by 1161 Cancers. And no, that gives us the tangent to be equal to negative 60 divided by 11. Now, what do we know about the rest of values? Well, for the core tangent, OK, the core tangent, the core tangent is the reciprocal of the tangent. OK. So since the tangent is negative 60 divided by 11, the core tangent must be negative 11 divided by 60. Next, we know that the C can is the reciprocal of the cosine function. So that's gonna be one divided by cost theta. OK? And since the cosine, the, since cosine theater is negative 11 divided by 61 that means the C count must be negative 61 divided by 11. Finally, for our course, you can OK, we know the co sequence is the reciprocal of the sine function. So that's the reciprocal of 60 divided by 61 which equals 61 divided by 60. Therefore, the sine of theta is 60 divided by 61. The core sine is negative 11 divided by 61. The tangent is negative 60 divided by 11. The core tangent is negative 11 divided by 60. The C count is negative 61 divided by 11. And finally, the core C count is 61 divided by 60. If we look back at our answer choices, that ac is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

A point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.
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Key Concepts

Unit Circle Definition

Trigonometric Functions on the Unit Circle

Sign of Trigonometric Functions in Quadrants

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