Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [trigonometry]

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

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles.

Trigonometry is most simply associated with planar right-angle triangles. The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles.

One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. Trigonometry on surfaces of negative curvature is part of hyperbolic geometry.

Trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, these are identities involving certain functions of one or more angles.

See Wikipedia's list of trigonometric identities.

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Rotating a rectangle

Lets say we have a rectangle with height h and width w. If we rotate it by d degrees, what would be the width and height of the window to display it without any clipping? I mean what is the formula to calculate wh and ww?
Majid Fouladpour
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Finding $\sin(\alpha+\beta)$ and $\cos(\alpha+\beta)$ if $\sin\alpha+\sin\beta=a$ and $\cos\alpha+\cos\beta=b$

If: $\sin(\alpha) + \sin(\beta) = a$ and $\cos(\alpha) + \cos(\beta) = b$ Determine: $\sin(\alpha + \beta) = ?$ and $\cos(\alpha + \beta) = ?$ The right answers: $$\sin(\alpha + \beta) = \frac{2ab}{a^2 + b^2} \qquad \cos(\alpha + \beta) =…
Dimitar Kapitanov
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Trig and de Moivre's theorem

A) Use de Moivre's theorem to prove that $\cos^4\theta = 8\cos^4\theta - 8\cos^2\theta + 1$ B) Therefore deduce that $\cos(\pi/8) = \left(\frac{2 + \sqrt{2}}{4}\right)^{1/2}$ C) and write down an expression for $\cos(3\pi/8)$. I have proved the…
1819023
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What's the importance of the trig angle formulas?

What's the importance of the trig angle formulas, like the sum and difference formulas, the double angle formula, and the half angle formula? I understand that they help us calculate some trig ratios without the aid of a calculator, but I guess I…
jrc03c
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Prove $(1-\cos x)/\sin x = \tan x/2$

Using double angle and compound angles formulae prove, $$ \frac{1-\cos x}{\sin x} = \tan\frac{x}{2} $$ Can someone please help me figure this question, I have no idea how to approach it?
Red Queen10101
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How to prove that $4 \cot ^{-1}\left(\sqrt{\phi }\right)+\cot ^{-1}\left(\frac{1}{4} \sqrt{22+17 \sqrt{5}}\right)=\pi$

Trying to answer this question (five years too late), I found to the surprising identity $$4 \cot ^{-1}\left(\sqrt{\phi }\right)+\cot ^{-1}\left(\frac{1}{4} \sqrt{22+17 \sqrt{5}}\right)=\pi$$ How to prove it ?
Claude Leibovici
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Why is the line between two points called the line of the "secant"?

The definition of the slope of the line of the secant is: slope = $\frac{y2-y1}{x2-x1}$ The definition of the slope of the tangent line is: $\lim_{h->0}\frac{f(x+h)-f(x)}{h}$ I understand why they call it the tangent line since the angle to the x…
Klik
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Given that $\;\sin^3x\sin3x = \sum^n_{m=0}C_m\cos mx\,,\; C_n \neq 0\;$ is an identity . Find the value of n.

Problem : Given that $\sin^3 x \sin 3x = \sum^n_{m=0}C_m \cos mx, C_n \neq 0 $ is an identity. Find the value of n. I tried : $\sin3x = 3\sin x - 4\sin^3 x$ but unable to reach to any point.... Please suggest further ....Thanks..
Sachin
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Value of $\sin(π/x)$ for $x$ positive integer

Is there a known Generalized algebraic formula for the following: $\sin(\pi/n)$ here, $n$ is an positive integer
GSA_1
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Proof using trigonometry that circle circumference is $2 \pi R$

Using trigonometry, I would like to prove that the circumference of a circle is $2\pi$ times its radius. Can someone help please?
user78259
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Another trigonometric proof...?

...sigh..another problem how shall I prove the following? $$ {\cot A\over1- \tan A} + {\tan A \over 1- \cot A} = 1 + \tan A + \cot A$$ so what now? the following's what I've done: $$\cot A - \cot^2 A + \tan A- \tan^2 A \over 2 - \tan A - \cot A$$
Ghost
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$A\cos(\theta) + B\sin(\theta)$ for complex $A,B$

Does the equation $$A\cos(\theta) + B\sin(\theta) = \sqrt{A^2+B^2}\cos(\theta + \gamma) \label{1} \tag{1}$$ with $\gamma = \arg(A-jB)$ require that $A$ and $B$ be real, or can they be complex? Consider the case $B= jA$ which results…
Dan Boschen
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Solving equation $\tan(5π\cos\alpha) = \cot(5π\sin\alpha)$

$$\tan(5π\cos \alpha) = \cot(5π\sin \alpha)$$ I did that $\tan(5π\cos\alpha) = \tan\left[\frac π2-5π\sin\alpha\right]$ And then used the solution of Trigonometric Equation $\tan(\theta)=\tan(\beta)$ Which is $\theta = nπ + \beta$, $n$ is an…
King
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Finding $\alpha$ such that $\tan\alpha = \frac{(1+\tan 1^{\circ})\cdot (1+\tan 2^{\circ})-2}{(1-\tan 1^{\circ})\cdot(1-\tan 2^{\circ})-2}$

I am preparing for my college entry test and I ran into this problem in my book: $$\tan\alpha = \frac{(1+\tan 1^{\circ})\cdot (1+\tan 2^{\circ})-2}{(1-\tan 1^{\circ})\cdot(1-\tan 2^{\circ})-2}$$ I should find the angle $\alpha$. Can anyone help me…
Radiant
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Generalizing trig-sum to product using complex exponentials

Consider, $$ \sin A \sin B$$ Using exponential definition of sine, $$ \frac{ e^{iA} - e^{-iA} }{2i} \cdot \frac{ e^{iB} - e^{-iB} }{2i}$$ $$ =\frac{1}{-4} ( e^{ i (A+B) } - e^{i (A-B)} -e^{ -i(A-B) } + e^{ - i(A+B)})$$ $$ =\frac{-1}{4} (…
tryst with freedom
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