High school trigonometry questions and answers

f(t)=sin(2t)

I calculated ${\displaystyle {\sin 75}^{\circ }}$ as ${\displaystyle \frac{1}{2\sqrt{2}}+\frac{\sqrt{3}}{2\sqrt{2}}}$, but the answer is ${\displaystyle \frac{\sqrt{2}+\sqrt{6}}{4}}$. What went wrong?
I calculated the exact value of ${\displaystyle {\sin 75}^{\circ }}$ as follows:
${\displaystyle {\sin 75}^{\circ }=\sin ({30}^{\circ }+{45}^{\circ })}$
$=\sin 30°\cos 45°+\cos 30°\sin 45°$
$=\frac{1}{2}·\frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2}·\frac{1}{\sqrt{2}}$
$=\frac{1}{2\sqrt{2}}+\frac{\sqrt{3}}{2\sqrt{2}}$
The actual answer is
${\displaystyle \frac{\sqrt{2}+\sqrt{6}}{4}}$

(ad≥ + 2ycost = coser?t

CA⃡ is the perpendicular bisector of̅BD̅̅̅̅.
Given that AB=3x and AD=4x-1, then what
is the value of x and the hypotenuse?

The area A of a triangle is given by bab sin 0, where a and b are the lengths of two sides and O is the angle between these sides. Suppose that a = 5, b = 10 and 0 = . (a) Find the rate at which A changes with respect to a if b and 0 are held constant. (b) Find the rate at which A changes with respect to 0 if a and b are held constant. (c) Find the rate at which b changes with respect to a if A and O are held constant.

assume a,b starts on the positive x-axis and moves around in a circle. determine a function of a segment from the origin to a point a,b

1. A Harris Interactive survey for InterContinental Hotels & Resorts asked respondents, “When traveling internationally, do you generally venture out on your own to experience culture, or stick with your tour group and itineraries?” The survey found that 23% of the respondents stick with their tour group (USA Today, January 21, 2004).
2. In a sample of six international travelers, what is the probability that two will stick with their tour group?
3. In a sample of six international travelers, what is the probability that at least two will stick with their tour group?
4. In a sample of 10 international travelers, what is the probability that none will stick with the tour group?

sinh(x-y)=Sinhx*coshy-sinhy**coshx

Let ${\displaystyle S_n=\sum _{j=1}^n\sin (\sqrt{j}\pi )}$.

Show $S_{(2M{)}^2}<0$ and $S_{(2M+1{)}^2}>0(M=1,2,3,...)$ .