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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Two different trigonometric identities giving two different solutions

Using two different sum-difference trigonometric identities gives two different results in a task where the choice of identity seemed unimportant. The task goes as following: Given $\cos 2x =-\frac {63}{65} $ , $\cos y=\frac {7} {\sqrt{130}}$,…
0lt
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Solving $\cos x+\sin x-1=0$

How does one solve this equation? $$\cos {x}+\sin {x}-1=0$$ I have no idea how to start it. Can anyone give me some hints? Is there an identity for $\cos{x}+\sin{x}$? Thanks in advance!
Mathxx
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Find an algebraic expression for sin(arccos(x))

I have a question that asks me to find an algebraic expression for sin(arccos(x)). From the lone example in the book I seen they're doing some multistep thing with the identities, but I'm just not even sure where to start here. It's supposed to be…
windy401
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How to prove a trigonometric identity $\tan(A)=\frac{\sin2A}{1+\cos 2A}$

Show that $$ \tan(A)=\frac{\sin2A}{1+\cos 2A} $$ I've tried a few methods, and it stumped my teacher.
Rollo Montgomery Konig-Brock
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Simplifying and evaluating $\cot 70^\circ+4\cos 70^\circ$

I have to simplify and evaluate this : $$\cot 70^\circ+4\cos 70^\circ$$ On evaluating it, the answer comes out to be $1.732$, or $\sqrt 3$ . I tried to get everything in $\sin$ and $\cos$, but it doesn't go any further. Any hints?
antman
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Prove that $ \cos x - \cos y = -2 \sin ( \frac{x-y}{2} ) \sin ( \frac{x+y}{2} ) $

Prove that $ \cos x - \cos y = -2 \sin \left( \frac{x-y}{2} \right) \sin \left( \frac{x+y}{2} \right) $ without knowing cos identity We don't know that $ \cos0 = 1 $ We don't know that $ \cos^2 x + \sin^2 x = 1 $ I have managed to prove it using the…
Gregory Peck
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Range of trigonometric functions

I would like to know if there is a simple approach to find the range of functions in the form: $$\sin x\sin2x$$ $$\cos x\cos3x$$ $$\sin 2x\cos 4x$$ For example, finding the range of a function in the form: $$a\cos\theta + b\sin\theta$$ is simple…
Gummy bears
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Evaluating a product of tangents

(1) Evaluation of $\displaystyle \tan \left(\frac{\pi}{7}\right)\cdot \tan \left(\frac{2\pi}{7}\right)\cdot \tan \left(\frac{3\pi}{7}\right) = $ (2) Evaluation of $\displaystyle \tan\left(\frac{\pi}{11}\right)\cdot…
juantheron
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Why doesn't $\arccos x = -\tfrac12\sqrt{3}$ have any solutions?

I have this exercise with an unclear answer. The question is this: $$\arccos x = -\frac{\sqrt3}{2}\,.$$ The answer is this: $$\begin{gather*} \varphi(x)= \arccos x\\ V_\varphi = [0,\pi]\\ -\frac{\sqrt3}{2}\notin V_\varphi\,. \end{gather*} $$ can…
Johan Hjalmarsson
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Find the side of an equilateral triangle given only the distance of an arbitrary point to its vertices

Triangle $ABC$ is an equilateral triangle and $P$ is an arbitrary point inside it. The distance from $P$ to $A$ is $4$ and the distance from $P$ to $B$ is $6$ and the distance from $P$ to $C$ is $5$. How to find the side of an equilateral triangle…
ward
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How to prove that $2 \arctan\sqrt{x} = \arcsin \frac{x-1}{x+1} + \frac{\pi}{2}$

I want to prove that$$2 \arctan\sqrt{x} = \arcsin \frac{x-1}{x+1} + \frac{\pi}{2}, x\geq 0$$ I have started from the arcsin part and I tried to end to the arctan one but I failed. Can anyone help me solve it?
tomas_haffner
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A strange trigonometric equation

Today,in our class, we received a trigonometric equation $$\sin^{10}{x}+\cos^{10}{x}=\frac{29}{16}\cos^4{2x}$$ and the question was to find the general solution of this equation. My approach was, at First, trying to show that there were no solutions…
Dinesh
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Double Angle Equations

$\cos2x=\frac1{\sqrt2}$ is the original problem, and I have to solve for $x$. However, I'm not sure what to do after I substitute the double angle formulas for $\cos2x$. I know that $\frac1{\sqrt2}$ can be rationalized to $\frac{\sqrt2}{2}$.
Min
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Finding the value of $\sin {\frac{31 \pi}3}$

The task was to find out the value of $$\sin\frac{31\pi}3$$ This is a example in my book in which following steps are shown: $$\begin{align} \sin\frac{31\pi}3& =\sin\left(10\pi +\frac\pi3\right)\\ &=\sin\frac\pi3\\ &=\frac{\sqrt3}2 \end{align}$$ I…
Freddy
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Does the Law of Sines and the Law of Cosines apply to all triangles?

Do the Law of Sines and the Law of Cosines apply to all triangles? Particularly, could you use these laws on right triangles? That is, could you use these laws instead of the Sine=opposite/hypotenuse, Cosine=adjacent/hypotenuse, and…
Learner
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