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Solve trigonometric equations I

The difference in length of a spring on a pogo stick from its non-compressed length when a teenager is jumping on it after θ\theta seconds can be described by the function y=2sin⁡(θ)−3y = 2\sin(\theta) - \sqrt{3}. Part A: Determine, algebraically, all values where the pogo stick's spring will be equal to its non-compressed length. Show all necessary calculations. (88 points) Part B: If the angle was doubled, that is θ\theta became 2θ2\theta, what are the solutions in the interval [0,2π)[0, 2\pi)? Solve algebraically and show all necessary calculations. How do these compare to the original function? (99 points) Part C: A toddler is jumping on another pogo stick whose length of their spring can be represented by the function y=−3cos⁡(2θ)y = -\sqrt{3} \cos(2\theta). At what times are the springs from the original pogo stick and the toddler's pogo stick lengths equal? Solve algebraically and show all necessary calculations. (88 points)
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For the given pair of equations, give the slopes of the lines, and then determine whether the two lines are parallel, perpendicular, or neither.\newline12x−12y=312x+16y=−9 \begin{array}{l} 12 x-12 y=3 \\ 12 x+16 y=-9 \end{array} \newlineSelect the correct choice below and, if necessary, fill in the answer box to complete your choice.\newlineA. The slope of 12x−12y=3 12 x-12 y=3 is □ \square .\newline(Type an integer or a simplified fraction.)\newlineB. The slope of 12x−12y=3 12 x-12 y=3 is undefined.\newlineSelect the correct choice below and, if necessary, fill in the answer box to complete your choice.\newlineA. The slope of 12x+16y=−9 12 x+16 y=-9 is □ \square .\newline(Type an integer or a simplified fraction.)\newlineB. The slope of 12x+16y=−9 12 x+16 y=-9 is undefined.\newlineThe lines are □ \square
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Find all solutions with −90∘<θ<90∘-90^\circ < \theta < 90^\circ. Give the exact answer(s) in simplest form. If there are multiple answers, separate them with commas.\newlinetan⁡(θ)=0\tan (\theta) = 0\newline____ ∘\_\_\_\_\,^\circ
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The table gives selected values of the differentiable function ff. Below is Leo's attempt to write a formal justification for the fact that there exists a value cc where a<c<ba < c < b such that f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}. Is Leo's justification complete? If not, why? Leo's justification: ff is differentiable on [a,b][a, b]. So, according to the mean value theorem, there exists a value cc somewhere between aa and bb such that f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}.
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Solve the SSA triangle. Indicate whether the given measurements result in no triangle, one triangle, or two triangles. Solve each resulting triangle. Round each answer to the nearest tenth. \newlineA=35∘a=24b=19 A=35^\circ \quad a=24 \quad b=19 \newlineSelect the correct choice below and, if necessary, fill in the answer box to complete your choice.\newlineA. There is only one possible solution for the triangle.\newlineThe measurements for the remaining angles \newlineB B and \newlineC C and side \newlinec c are as follows.\newlineB=□∘ B=\square^\circ \newlineC=□∘ C=\square^\circ \newline(Round to the nearest tenth as needed.)\newlinec=□ c=\square \newlineB. There are two possible solutions for the triangle.\newlineThe measurements for the solution with the longer side \newlinec c are as follows.\newlineB1=□∘ B_{1}=\square^\circ \newlineC1=□∘ C_{1}=\square^\circ \newlineB B 00\newlineThe measurements for the solution with the shorter side \newlinec c are as follows.\newlineB B 22\newlineB B 33\newline(Round to the nearest tenth as needed.)\newlineC. There are no possible solutions for the triangle.\newlineView an example\newlineGet more help -\newlineClear all
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Kajal tried to solve the differential equation dydx=−x2y2 \frac{d y}{d x}=-x^{2} y^{2} . This is her work:\newlinedydx=−x2y2 \frac{d y}{d x}=-x^{2} y^{2} \newlineStep 11: ∫−y−2dy=∫x2dx \quad \int-y^{-2} d y=\int x^{2} d x \newlineStep 22: y−1=x33+C\quad y^{-1}=\frac{x^{3}}{3}+C \newlineStep 33: y=3x3+C\quad y=\frac{3}{x^{3}}+C \newlineIs Kajal's work correct? If not, what is her mistake?\newlineChoose 11 answer:\newline(A) Kajal's work is correct.\newline(B) Step 11 is incorrect. The separation of variables wasn't done correctly.\newline(C) Step 22 is incorrect. Kajal didn't integrate x2 x^{2} correctly.\newline(D) Step 33 is incorrect. Kajal didn't take the reciprocal of x33+C \frac{x^{3}}{3}+C correctly.
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