Mathematics Stack Exchange
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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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If $t = \tan (x/2)$, find expressions for $\sin x, \cos x$ in terms of $t$. Hence, solve the equation $3\sin x - 4\cos x = 2$.

If $$t = \tan \frac{x}{2},$$ find expressions for $\sin x, \cos x$ in terms of $t$. Hence, solve the equation $$3\sin x - 4\cos x = 2.$$ Attempt: I have been solving a lot of trig questions lately but this is different. I don't know how to…
user274246
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How to simplify $\sin^4 x+\cos^4 x$ using trigonometrical identities?

$\sin^{4}x+\cos^{4}x$ I should rewrite this expression into a new form to plot the function. \begin{align} & = (\sin^2x)(\sin^2x) - (\cos^2x)(\cos^2x) \\ & = (\sin^2x)^2 - (\cos^2x)^2 \\ & = (\sin^2x - \cos^2x)(\sin^2x + \cos^2x) \\ & = (\sin^2x -…
Zauberkerl
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trigonometry equation, $[\cos(2x)]^2-\sin(2x)=1$

I tried to solve this equation: $[\cos(2x)]^2-\sin(2x)=1$ I got different answer from the book. the answer in the book: $x=-45+180k$ , $x=90k$ Am I right? $x = -45+180k$ is equal to $x= 135+180k$ ? How can I check if it's the same? Or maybe I did a…
Silas2033
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Prove that $-\sqrt{2}\leq \sin\theta+\cos\theta\leq\sqrt{2}$

We have to do it without calculus or any complex inequality. Level of complexity is that we cannot even use the AM-GM inequality. So I tried,…
Aditya Agarwal
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Prove $\sin^2 \theta +\cos^4 \theta =\cos^2 \theta +\sin^4 \theta $

Prove $$\sin^2(\theta)+\cos^4(\theta)=\cos^2(\theta)+\sin^4(\theta)$$ I only know how to solve using factoring and the basic trig identities, I do not know reduction or anything of the sort, please prove using the basic trigonometric identities and…
Sunny Mann
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Solve $\cos \frac{4x}{3}=\cos x+1$

Solve the equation \begin{equation} \cos \frac{4x}{3}=\cos x+1\tag 1\end{equation} I had tried by taking $\cos\dfrac x3=t$ and from this we have $\displaystyle\cos\frac{4x}3=2\left(2t^2-1\right)^2-1; \cos x=4t^3-3t$ $(1) \iff…
mja
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How to prove the following trigonometric equation

Possible Duplicate: Sum of the reciprocal of sine squared Can anyone help me? If $\alpha=\frac{\pi}{N}$, where $N\geqslant 2$ How to prove the following equation? $$\sum_{k=1}^{N-1}\frac1{\sin^2(k\alpha)}=\frac{N^2-1}{3}$$
staper
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$a^2+b^2=2Rc$,where $R$ is the circumradius of the triangle.Then prove that $ABC$ is a right triangle

If in a triangle $ABC$,$c$ is the longest side and $a^2+b^2=2Rc$,where $R$ is the circumradius of the triangle.Then prove that $ABC$ is a right triangle. $a^2+b^2=2Rc\Rightarrow a^2+b^2=\frac{c^2}{\sin C}$ $\sin C=\frac{c^2}{a^2+b^2}$,how to proceed…
diya
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The complete solution set of $[\sin^{-1}x]>[\cos^{-1}x]$ is

The complete solution set of $[\sin^{-1}x]>[\cos^{-1}x]$ is $(A)[\sin1,1]\hspace{1 cm}(B)[\frac{1}{\sqrt2},1]\hspace{1 cm}(C)(\cos 1,\sin 1)\hspace{1 cm}(D)[0,1]\hspace{1 cm}$ I think its answer should be (B) as $\sin^{-1}x$ and $\cos^{-1}x$ meet at…
Brahmagupta
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If $\cos(\alpha-\beta)+\cos(\beta-\gamma)+\cos(\gamma-\alpha)+1=0$,show that $\alpha-\beta$ or $\beta-\gamma$ or $\gamma-\alpha$ is multiple of $\pi$.

This question is from SL Loney. If $\cos(\alpha-\beta)+\cos(\beta-\gamma)+\cos(\gamma-\alpha)+1=0$, then show that $\alpha-\beta$ or $\beta-\gamma$ or $\gamma-\alpha$ is a multiple of $\pi$. My try: Let $\alpha-\beta=A$, $\beta-\gamma=B$,…
Vinod Kumar Punia
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How to prove an identity (Trigonometry Angles--Pi/13)

In this page http://mathworld.wolfram.com/TrigonometryAnglesPi13.html I found equation (11) and…
Bless
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Infinite limit of trigonometric series

The value of $\displaystyle\lim_{n\to\infty}(\sin^4x+\frac{1}{4}\sin^4(2x)+\cdots+\frac{1}{4^n}\sin^4(2^nx))$ is (A) $\sin^4x$ (B) $\sin^2x$ (C) $\cos^2x$ (D) does not exist My…
Vinod Kumar Punia
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Fundamental definition of trigonometric functions

What is the most basic or fundamental definition of a trigonometric function, (say sine)? How is sine of an angle defined? I looked up on wikipedia, and it seems that sine of an angle stems from this definition: In a right triangle, the sine of an…
shobhu
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The exact value of $\cot\frac{7\pi}{6}$?

I am working on a trigonometry question at the moment and am very stuck. I have looked through all the tips to solving it and I cant seem to come up with the right answer. The problem is What is exact value of $$\cot \left(\frac{7\pi}{6}\right)?…
Sarah
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Solving $6 \cos x - 5 \sin x = 8$

My attempt: Using the formula for linear combinations of sine and cosine: $$A \cos x+B \sin x=C \sin (x+\phi)$$ $$ \sqrt{51} \left(\frac{6}{\sqrt{51}} \cos x - \frac{5}{\sqrt{51}}\sin x\right) = 8 $$ $$ \frac{6}{\sqrt{51}} \cos x -…
Evgeny Semyonov
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