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.

29665 questions
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Finding side length and circumradius in a triangle

in a triangle $ABC$ if $2\sin B= \sin A $ and $a,b,c$ be the length of side $BC,CA$ and $AB$ and length of internal angle bisector through $A$ is $\displaystyle \frac{\sqrt{2}}{3}$ unit and the equation $25\cos^2(A-B)+x^2-40\cos(A-B)-2x+17=0$ has…
DXT
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Solving for $y$ with $\arctan$

I know this is a very low level question, but I honestly can't remember how this is done. I want to solve for y with this: $$ x = 2.0 \cdot \arctan\left(\frac{\sqrt{y}}{\sqrt{1 - y}}\right) $$ And I thought I could do…
RileyE
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Given $3\cos x - 4 \sin x = 2$, find $3 \sin x + 4 \cos x$ without first solving for $x$

If $$3\cos{x}-4\sin{x}=2$$ find $$3\sin{x} +4\cos{x} $$ I have solved the equation for $x$, then calculated the required value, but I think there is a direct solution without solving the equation.
Hussien Mohamed
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Solving $\cos x + \cos 2x - \cos 3x = 1$ with the substitution $z = \cos x + i \sin x$

I need to solve $$\cos x+\cos 2x-\cos 3x=1$$ using the substitution$$z= \cos x + i \sin x $$ I fiddled around with the first equation using the double angle formula and addition formula to get $$\cos^2 x+4 \sin^2x\cos x-\sin^2 x=1$$ which gets me…
Yep
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How to solve the trigonometric inequality $\csc(2x) \leq \sec(x + \frac{\pi}{6})$ in $\mathbb{R}$?

I have difficulty in solving the inequality $\csc(2x) \leq \sec(x + \frac{\pi}{6})$ in $\mathbb{R}$. First, I need to solve the equation $\csc(2x) - \sec(x + \frac{\pi}{6}) = 0$. I see that the period is $2\pi$. My problem is that I have no idea…
user513928
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Prove $\tan(\frac{\alpha}{2})\tan(\frac{\beta}{2})=\frac{1}{5}$

Given $2\sin(\alpha)+2\sin(\beta)=3\sin(\alpha+\beta)$, prove that $\tan(\frac{\alpha}{2})\tan(\frac{\beta}{2})=\frac{1}{5}$ Also we know that all the expressions are different from zero and defined. Including the expressions we received during the…
TuYu
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Proving a tangent value equality

Show that $\tan50 \tan60 \tan70 = \tan80$. I have used compound angle formula for tan to bring $\tan 10$ into it as $\frac{1}{\tan 10}=\tan 80$, but I can't seem to get it to come out.
RedG
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Superposition of two waves

Two waves on the string of a musical instrument superposition to give you a third wave. One wave is given by the equation $y =\sin(30t)$ and the other is given by $y = \sin(32t)$. The superposition of the two waves makes a sound (a hum) of a…
Cookie
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Trigonometric equation result differs from given

i've got an equation: $$\sin^6(x) + \cos^6(x) = 0.25$$ and i'm trying to solve it using the sum of cubes formula, like this: $$ (\sin^2(x))^3 + (\cos^2(x))^3 = 0.25 $$ $$ (\sin^2(x) + \cos^2(x))^2 (\sin^4(x) - \sin^2(x)\cos^2(x) + \cos^4(x)) = 0.25…
Dmitrii
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prove that $\displaystyle A\cdot B= 2^{-6}\bigg(\csc \frac{\pi}{22}-1\bigg)$

If $\displaystyle A= \prod^{5}_{k=1}\cos \left(\frac{k\pi}{11}\right)$ and $\displaystyle B= \sum^{5}_{k=1}\cos \left(\frac{k\pi}{11}\right)$ then show that $\displaystyle A\cdot B= 2^{-6}\bigg(\csc \frac{\pi}{22}-1\bigg)$ Attempt: $\displaystyle…
DXT
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Trigonometry Radians help

I have solutions to two problems, but I just have a question about these solutions. So I am to solve the problem in the image above. For b iv), The ratios I got is sin=-5/13, cos=-12/13, and tan=5/12. The angle I got is 22.6, but because the point…
ernest
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How many different tangent angles create reducible expressions like $\tan(60^\circ)=\sqrt3$?

How many different tangent angles create reducible expressions like $\tan(60^\circ)=\sqrt3$? Please include obscure angles. Where would information like this be found? There are in infinite number of degree steps to be considered. How to address…
User3910
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Solving $a \sin\theta + b \cos\theta = c$

Could someone help me with the steps for solving the below equation $$a \sin\theta + b \cos\theta = c$$ I know that the solution is $$\theta = \tan^{-1} \frac{c}{^+_-\sqrt{a^2 + b^2 - c^2}} - \tan^{-1} \frac{a}{b} $$ I just can't figure out the…
varun
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Can every finite product of cosines be rewritten as a finite sum of cosines?

I used TrigReduce on products of cosines in Mathematica. It led to me to develop the formula $\prod_{i=1}^k\cos\left(\beta_i\right)=\sum_{j=1}^{2^{k-1}}\frac{\cos\left(P_j\right)}{2^{k-1}}, k \geq1$ where $P_j$ is a permutation of the signed sums of…
user477818
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Trigonometric equation

I have been trying to solve this equation for over a week now: $$\tan5x-2\tan3x=\tan3x\tan5x$$ I found one solution $x=k\pi$ but I cannot prove that this is the only solution. It is equivalent to: $$\sin 5x \cos 3x - 2 \sin 3x \cos 5x = \sin 3x…
Adam
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