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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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Find the domain of a trig function

Find the Domain of the function $y=\sqrt{1+2 \sin x}$. My solution: $1+2 \sin x \geq 0$ (since square-root of a negative number is not defined) $2 \sin x \geq -1$ $\sin x \geq -1/2$ $x \ge -\frac{\pi}{6}$ $x \geq (2n-\frac16) \pi$; but the answer…
Vikram
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Two trigonometric eliminations

I have a couple of trigonometric elimination questions for you... $1) a \sin \theta = b \sin (2 \theta)$; $c \cos \theta = d \cos (2 \theta)$ *NOTE: the first equation was incorrect...I had $\cos 2 \theta$ and it should be $\sin 2 \theta$. I was…
bjcolby15
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What is arcsec (-2)?

The question asks to solve $\operatorname{arcsec}(-2) $. Options are a) π/3 b) −(π/3) c) 5π/3 d) 4π/3 e) 2π/3 So my thoughts are: this is quadrant 2 or 3; on the unit circle, $ x = -1/2 $. $\cos(t) = 1/2$ where $t$ is 60°. In quadrant 2 that is…
Marty B.
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Prove that $A + B = C$

I drew the diagram here I honestly do not see how $A$ and $B$ could possibly equal $C$.
user116528
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How do I solve $\sin^2 x=\cos x$?

I'm trying to solve a trigonometric equation, but I'm a bit stuck. The equation is this: $\sin^2 x = \cos x$ So far what I've done looks like this: $\sin^2 x - \cos x = 0$ $ (1 - \cos^2 x) - \cos x = 0$ $-\cos^2 x - \cos x + 1 = 0$ But from…
Threethumb
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what is the value of $\tan\theta$ and $\csc\theta$?

Please help me to solve this problem.I can not understand what to do. If $(a^2-b^2)\sin\theta + 2ab\cos\theta = a^2+b^2$ and $\theta$ is acute and positive angle then what is the value of $\tan\theta$ and $\csc\theta$ ?
Fazla Rabbi Mashrur
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What is the maximum value of this trigonometric expression

What is the maximum value of the expression $1/(\sin^2 \theta + 3\sin\theta \cos\theta+ 5\cos^2 \theta$). I tried reducing the expression to $1/(1 + 3\sin\theta$ $\cos\theta + 4\cos^2 \theta)$. How do I proceed from here?
Ajay George
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How prove this $\alpha+\beta+\gamma=n\pi$

let $\theta\in R$,and $\alpha\neq\beta\neq\gamma$ and such $$\dfrac{\cos{(\alpha+\theta)}}{\sin^3{\alpha}}=\dfrac{\cos{(\beta+\theta)}}{\sin^3{\beta}}=\dfrac{\cos{(\gamma+\theta)}}{\sin^3{\gamma}}$$ prove $$\alpha+\beta+\gamma=n\pi$$ My try:…
math110
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How find this $\sum_{i=1}^{5}\tan^4{\frac{i\pi}{11}}$

show that:$$\tan^4{\dfrac{\pi}{11}}+\tan^4{\dfrac{2\pi}{11}}+\tan^4{\dfrac{3\pi}{11}}+\tan^4{\dfrac{4\pi}{11}}+\tan^4{\dfrac{5\pi}{11}}=2365$$ my try: I think first we can find this…
user94270
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What is the maximum value of $4(\sin x)^2 + 3(\cos x)^2$

The question is: What is the maximum value of: $4\sin^2\theta + 3\cos^2\theta$ This is the way I did it: $4\sin^2\theta + 3\cos^2\theta = \sin^2\theta + 3\sin^2\theta + 3\cos^2\theta = \sin^2\theta + 3$ The max value of $\sin^2\theta$ is $1$, so the…
Shaurya Gupta
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How solve this equation

Solve the equation $$\cos \left(x+30^{\circ}\right)+\cos \left(x+10^{\circ}\right)=\cos \left(2x+10^{\circ}\right)+\cos 10^{\circ}$$ , where $x\in (0,\pi )$ My try:…
math110
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How find the value of $\cos{x}+\cos{y}$

let $$\dfrac{\cos{x}\cos{\dfrac{y}{2}}}{\cos{(x-\dfrac{y}{2})}}+\dfrac{\cos{y}\cos{\dfrac{x}{2}}}{\cos{(y-\dfrac{x}{2})}}=1$$ Find the value $$cos{x}+\cos{y}=?$$ this following My ugly solution:…
math110
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Solving a trigonometric system of equations

I am a math tutor at the local university and encountered a trigonometric problem which stumped me. (Of course, I'm certain that this material was covered in my students class, but they still didn't know how to solve it, either.) Basically, the…
Code-Guru
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Trigonometry : Simplify and find the value of $\tan\theta(1-\sec\frac{\theta}{2})(1-\sec\theta)(1-\sec2\theta)\dots(1-\sec2^{n-1}\theta)$ at n =1,2,3

Problem : Simplify and find the value of $\tan\theta(1-\sec\frac{\theta}{2})(1-\sec\theta)(1-\sec2\theta)\dots(1-\sec2^{n-1}\theta)$ at n =1,2,3 My approach :…
Sachin
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Irrationality of a unique positive root of $\sin{x} = x^2$

The equation $\sin{x} = x^2$ has a unique positive real root. I wonder if there is any standard technique how to show that this number is irrational (rational), preferably a technique which works also in other similar scenarios. I tried an inverse…
Adam
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