Evaluate each expression. See Example 4. cot² 135° - sin 30° +...

Question

Evaluate each expression. See Example 4. cot² 135° - sin 30° + 4 tan 45°

Accepted Answer

Welcome back. I am so glad you're here. We are asked to evaluate the following trigonometric expression. We have seven tangent degrees plus two cosine 60 degrees minus cotangent squared, 225 degrees. Our answer choices are answer choice, A seven answer choice B nine, answer choice C six and answer choice D eight. All right. Looking at these angles, we can see that we've got some special angles here that 45 degrees and the 60 degrees. And it's possible that looking at cotangent of 225 degrees, we might be able to use reference angles and have another special angle. Those are great. We have exact values for those special angles if you recall from previous lessons and from the tables in the textbook for the tangent of 45 degrees, that's equivalent to the exact value of one. So here we've got seven being multiplied by one. When we say seven tangent 45 degrees. Looking at those same charts, the cosine of 60 degrees that's equal to the exact value of one half. So that's plus two multiplied by one half. Now, for this last one, we have minus and for the cotangent of 225 degrees, let's take a look at a coordinate plane, we have a vertical Y axis and a horizontal X axis. They come together at the origin in the middle for 225 degrees. That's got its initial side along the positive part of the X axis. Then we head counterclockwise all the way into quadrant three where we have 225 degrees extended toward negative infinity for both X and Y. Now, if we want to know the reference angle for 225 degrees, that quadrant three angle, we're going to take the 225 degrees and subtract 180 degrees and that's going to give us 45 degrees. So our reference angle here is 45 degrees in quadruple one and 45 degrees. We can look at the cotangent of 45 degrees in that chart from the textbook and that's going to be equal to positive one. But we need to note a couple of things here. First, we're talking about cotangent in quadrant three is cotangent positive or negative. In quadrant three, we recall from previous lessons, we have that Pneumonic to help us remember where trigonometric functions are positive starting in the first quadrant. That's up to the right of the origin. They are all positive up into the left. In the second quadrant, we have sign. So a sine function and its reciprocal function are positive. And now in third quadrant where we're talking about A S and then we have T for the third quadrant. So tangent and its reciprocal function cotangent are positive in the third quadrant. So this one remains positive. Now, the other note here is that our cotangent function is being squared. So we need to be sure to square this one. Now, it's just a matter of simplifying seven multiplied by one is seven plus two, multiplied by one half. Two, multiplied by one half is one minus one squared. One squared is one. So we have seven plus one minus one and that equals seven. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Evaluate each expression. See Example 4. cot² 135° - sin 30° + 4 tan 45°

Key Concepts

Trigonometric Function Values at Special Angles

Cotangent and Its Relationship to Tangent

Order of Operations in Trigonometric Expressions

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