Tagged Questions

Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.

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Condition for trigonometric inequality

I want to prove the following statement: Suppose $\frac{1}{4}(\cos(\theta_1)+\cos(\theta_2))^2+\lambda^2(a\sin(\theta_1)+b\sin(\theta_2))^2\leq 1$ holds for all $\theta_1,\theta_2\in[-\pi,\pi]$, then ...
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Why is the sine and cosine always between $-1$ and $1$?

Why is the sine and cosine always between $-1$ and $1$? If I would have circle with a radius other than $1$, then it wouldn't be between $-1$ and $1$ anymore, would it? This also ties in with another ...
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Proof of $\arcsin x \le 2\arctan x$?

I am looking for a proof for the following 'fact': $$\arcsin x \le 2\arctan x \quad \forall x\in[0,1).$$ I put fact between single quotes, as the only proof I found is a plot by wolframalpha. I know ...
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How is $\tan^{-1}(a/b) = \tan^{-1}(a) - \tan^{-1}(b)$?

I'm having a problem proving: $\tan^{-1}(a/b) = \tan^{-1}(a) - \tan^{-1}(b)$ Thanks!
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The value of $\int_0^{2\pi}\cos^{2n}(x)$ and its limit as $n\to\infty$

Calculate $I_{n}=\int\limits_{0}^{2\pi} \cos^{2n}(x)\,{\rm d}x$ and show that $\lim_{n\rightarrow \infty} I_{n}=0$ Should I separate $\cos^{2n}$ or I should try express it in Fourier series?
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Calculating originally arc approximated by cubic bezier curve

I have an cubic bezier curve, which is representing an arc by an approximation. The approximation was calculated with the kappa constant: $$\\k = \frac43*(\sqrt{2}-1)$$ This means, that the ...
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Cyclic quadrilaterals - finding the size of an angle

I know this might seem like a really simple question, but I really don't understand where I am going wrong. I am familiar with cyclic quadrilaterals as well as their properties, but this question ...
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Finding Y coordinate of third triangle point when X coordinate and two other points are already known

Suppose you know the coordinates for points A and B of a triangle. We can refer to those coordinates as (Ay,Ax) and (By,Bx). Also, suppose you know the X coordinate for point C (Cx) but do not know ...
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Find all of the exact solutions of the equation and then list those solutions which are in the interval [0, 2pi)

This is for trigonometric equations and inequalities: Find all of the exact solutions of the equation and then list those solutions which are in the interval [0, 2pi) Cos(9x)=9
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Finding Trig Functions

I am given $$\cos(\beta) = \frac{\sqrt{15}}{8}$$ and I am asked to find the indicated trig function of $$\sin(90^{\circ}- \beta)$$ I know $$\sin(\beta)= \frac{7}{8}$$ but I don't know how to go about ...
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Limit as x approaches 0 from the left: $\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$

Help me find the limit as x approaches 0 from the left: $$\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$$ Thanks,
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Prove $\lim_{x\rightarrow 0}\cos (x)=1$ with the epsilon-delta definition of limits

Prove $$\lim_{x\rightarrow 0}\cos (x)=1$$ with the epsilon-delta definition of limits
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Moving a point around a circle

we're currently working on a game which involves a character that rotates around a point. We are using a rotation matrix to rotate a given a point (x,y) around another point by first translating to ...
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Using $x=\tan \theta$ to solve $\int x\sqrt{1+x^2}\,\mathrm dx$

I'm having a lot, I repeat, a lot of trouble with Calculus II, particularly trigonometric substitution. At the moment, I'm extremely confused as to how to integrate $\int x\sqrt{1+x^2}\,\mathrm dx$ ...
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