Jennifer H.

A current high school math teacher; tutor for 6 years

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Pre-Calculus

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Question:

Find all solutions to the equation $$\sin \theta= \sqrt{3}\cos\theta$$ for $$0 \leq \theta< 2\pi$$. Leave your answer in exact form.

Jennifer H.

Answer:

When solving for $$\theta$$ with two trig functions, it's best to first try and find a way to combine into one trig function, so we can more easily solve. In this equation, let's divide both sides by $$\cos\theta.$$ So we have: $$\sin \theta= \sqrt{3}\cos\theta$$ $$\frac{\sin \theta}{\cos\theta}= \sqrt{3}$$ Now, we recall that $$\frac{\sin\theta}{\cos\theta}= \tan\theta$$. So we can substitute $$\tan\theta$$ into the left hand side of the equation. This results: $$\tan\theta = \sqrt{3}$$ Now we need to think when is the tangent of an angle equal to $$\sqrt{3}$$? If the question did not specify to leave the answer in exact form, we could take the inverse tangent of both sides and get an approximate answer on our calculator. However, since we want an exact answer the easiest way is to visualize the unit circle. Recall that the $$\sin\theta$$ is the y coordinate on the unit circle, and the $$\cos\theta$$ is the x coordinate on the unit circle. Since $$\tan\theta=\frac{\sin\theta}{\cos\theta}$$, we are looking for the angle whose y coordinate divided by the x coordinate equals $$\sqrt{3}$$. This occurs at $$60$$ degrees or $$\frac{\pi}{3}$$ radians. However, since we are looking for all solutions between $$0 \leq \theta< 2\pi$$, we must glance around the entire unit circle to see if there are any other solutions. Notice this also occurs at $$240$$ degrees or $$\frac{4\pi}{3}$$ radians.

Geometry

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Question:

Consider two similar triangles, A and B. Triangle A has sides lengths of 3,4,5. Triangle B has side lengths of 6,8 and x. What value for x would make Triangle A and Triangle B similar?

Jennifer H.

Answer:

Determining if two shapes are similar is a very common question in Geometry. When answering questions like this, it's important to first think about what is given in the problem, and what else we need to know in order to properly answer. In this question, we are given no information about the angle values in either triangle. So, we can rule out any form of similarity by AA (angle-angle congruence). However, we do know information about the side lengths. Looking at Triangle A with sides 3,4,5, and Triangle B with side lengths of 6,8, and x. We can tell that side B appears to have side lengths exactly double that of Triangle A. That means, Triangle B is twice as big. Or, in "math terms", we say, Triangle B is a dilation of Triangle A with a scale factor of 2. Continuing this same dilation factor, the third side of Triangle B, x, must also be twice as big as the third side of Triangle A, 5. This means that if x=10, Triangle A and Triangle B are similar under SSS, as Triangle B is a dilation with a scale factor of 2 from Triangle A.

Algebra

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Question:

Find the equation of the line that passes through the point (4,5) and (-3,2).

Jennifer H.

Answer:

Remember that to find the equation of a line, we need two things: 1.) The slope 2.) The y-intercept (where the line hits the y-axis) Let's first start by finding the slope. Remember we calculate the slope of a line by using the equation $$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$, where $$m$$ is the variable used for slope. (Recall that this formula simply means the difference in your vertical distance divided by the distance in the horizontal distance-- like finding the steepness of a a stair-step!) Let's use this formula with our points (4,5) and (-3,2). $$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$ $$m=\frac{2-5}{-3-4}$$ $$m=\frac{-3}{-7}$$ Therefore, we can conclude that the slope of this line is $$\frac{-3}{-7}=\frac{3}{7}$$. Remember that the equation of a line looks like y=mx+b. And so far, we have figured out the slope, m. Next we need to find the y-intercept, b. To figure out when this particular line crosses the y-axis, we can plug in a coordinate (x,y) and our slope m into the equation and solve for b. Although we can use either coordinate, (4,5) or (-3,2), let's choose the first one (4,5) for our x and y coordinate. Next we will use our value of $$\frac{3}{7}$$ as m. Plugging those numbers into the equation we get: $$5=\frac{3}{7}(4)+b$$ $$5= \frac{12}{7}+b$$ $$5-\frac{12}{7}=b$$ $$\frac{60}{5}-\frac{12}{5}=b$$ $$\frac{48}{5}=b$$ Now that we have found our y-intercept, b, we can write our equation of the line as: $$y=\frac{3}{7}x+\frac{48}{5}$$.

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