In Exercises 7–14, use the given information to find the exact...

Question

In Exercises 7–14, use the given information to find the exact value of each of the following: c. tan 2θ 15 sin θ = -------- , θ lies in quadrant II. 17

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression using the provided data. The expression we want to find the value of is the tangent of two beta. We are told that the sign of beta equals 12 37th and that beta lies in quadrant two, we are given four answer choices. So to begin with, I want to find my double angle identity for the tangent of two beta. So that is that the tangent of two beta is equal to two multiplied by the tangent of beta divided by one minus the tangent squared of beta. So we definitely need to find what is the tangent of beta to solve this problem? We are given that the sign of beta is 12 37th. So from this, I can recall that sign is the same as the ratio of Y to R when we're working with a coordinate plane. And from there, I recall that the tangent of beta would be the ratio of Y to X. So right now, I know that R equals 37 Y equals 12, but I don't know what X is yet. Luckily R X and Y are based on the Pythagorean theorem. So I can use X squared plus Y squared equals R squared. And the values I know for Y and R to find the value of X. So I know that X squared plus 12 squared has to equal 37 squared. So solving for the X variable, if X squared plus 144 would equal 1369 to get the X squared by itself, I subtract 144 from both sides. So X squared equals 1225. Recall the opposite of a square is a square root. And when I take the square root of both sides, I get the X equals plus or minus 35. Since we are told that this point or that beta lies in quadrant two. I know that in quadrant two X values are negative because they're to the left of the Y axis. So X has to equal negative 35. Now that I know that X equals negative 35. And I already know the value for Y, I can plug them in to find that the tangent of beta equals 12 over negative or negative 12 35th. And I can use this to find my double angle. So plugging the negative 12 35th into the double angle for identity. So we have the tangent of two beta will equal two, multiplied by the tangent of beta. So negative 12 35th divided by one minus our tangent of beta squared. So negative 12 35th squared in the numerator, my multiplication will give us negative 35th. In the denominator, I have one minus. And when I do negative 12 35th squared, I will get a positive 144 oh um divided by 1225. And I want to do the subtraction in my denominator before I do anything else. So my numerator stays the same. I have negative 24 35th. And when I do one minus divided by 1225 I get 1081 divided by 1225. I'm going to rewrite this as a fraction sentence instead of a complex fraction. So I have negative 24 divided by divided by 1081 divided by 1225. We're calling my rules of fractions. I know this is the same as multiplying by a reciprocal. So I have negative 24 divided by multiplied by divided by 1081. I want to reduce this if I can and I can cross reduce 1225. It looks like 35 goes into 35 once and goes into 1225 35 times. So when I multiply straight across my numerators, negative 24 multiplied by 35 gives me negative four hun I'm sorry, negative 840 and one multiplied by 1081 is 1081. So the tangent of two beta would be negative 840 divided by 1081 which is answer choice D have a nice day.

In Exercises 7–14, use the given information to find the exact value of each of the following: c. tan 2θ 15 sin θ = -------- , θ lies in quadrant II. 17

Key Concepts

Trigonometric Ratios and Quadrants

Double-Angle Identity for Tangent

Finding Cosine and Tangent from Sine

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