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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prove that : $\cos x \cdot \cos(x-60^{\circ}) \cdot \cos(x+60^{\circ})= \frac14 \cos3x$

I should prove this trigonometric identity. I think I should get to this point : $\cos(3x) = 4\cos^3 x - 3\cos x $ But I don't have any idea how to do it (I tried solving $\cos(x+60^{\circ})\cos(x-60^{\circ})$ but I got nothing)
Triomath
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Derivation in simple harmonic motion

I'm having trouble seeing the equality used in Wikipedia's article on simple harmonic motion. $c_1 \cos(\omega t) + c_2 \sin(\omega t) = A\cos(\omega t- \varphi)$ where $A = \sqrt{c_1^2 + c_2^2}$ and $\tan \varphi = c_2/c_1$, $\omega$ is a constant,…
user098123
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Find point of right triangle by hypotenuse and another point

Is it possible to find the coordinates of the point marked (?,?) if I have a rectangle/right triangle with a given point and the length of the hypotenuse? See image: Thanks everyone.
Michael Seltenreich
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Evaluating solutions for a given trigonometric equation over a specified interval

This question is similar in form to this one: Finding all Trigonometric Solutions of an Equation within a Given Interval However, I want to verify my method of solving, as it would appear I have made some logical error in my processes. My method…
user146046
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Simplify $\arctan (\frac{1}{2}\tan (2A)) + \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) $

How to simplify $$\arctan \left(\frac{1}{2}\tan (2A)\right) + \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) $$ for $0< A< \pi /4$? This is one of the problems in a book I'm using. It is actually an objective question , with 4 options given , so i…
A Googler
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What does $\sin^{2k}\theta+\cos^{2k}\theta=$?

What is the sum $\sin^{2k}\theta+\cos^{2k}\theta$ equal to? Besides Mathematical Induction,more solutions are desired.
yibotg
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Basic Trigonometry: Why is the smaller side = $\sin(\theta)$

I am learning the proof for the Special Trigonometric Limit. $\lim_{x \to 0} \dfrac{\sin x}{x} = 1$ But I have a basic Geometry question I can't figure out. Why is the side of the smaller triangle = $\sin(\theta)$? (See the marker highlighted side).…
George
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unit circle, derive number for any degree, cosinus and sinus

$\sin(90°)= \sin(\frac{1}{2}\pi)= 0$ $\cos(90°)= \cos(\frac{1}{2}\pi)= 1$ $\sin(60°)= \sin(\frac{1}{3}\pi)=\frac{\sqrt{3}}{2}$ $\cos(60°)= \cos(\frac{1}{3}\pi)=\frac{1}{2} $ $\sin(45°)= \sin(\frac{1}{4}\pi)=\frac{\sqrt{2}}{2}$ $\cos(45°)=…
kiltek
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How can this trigonometrics equation be solved exactly, if possible?

I was working on an approximation for the sine function, in which I needed to calculate the maximum error to work on a compensation polynomial. My approximation was this: $$f(x) = \frac {4} {\pi^2} x (\pi - |x|)$$ Then, obviously the error is…
orlp
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Confusion to use formula $l = r\theta$

I have been teaching my brother some trignometry. There is a formula as arc length of circumference of a circle. The basic formula is $$l = r\theta.$$ But sometimes for length they use $l = 2r$ and other times $l = 2\pi r$. I want to know when to…
Muhammad Taqi
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How to evaluate $\cos(22^\circ)\cos(38^\circ) - \sin(22^\circ)\sin(38^\circ)$?

How does one evaluate this? Does this generalize to $\cos(x)\cos(y) - \sin(x)\sin(y)$?
Flack Ninja
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How to solve $\tan x =\sin(x+45^{\circ})$?

How do I solve $\tan x = \sin(x +45^{\circ})$? This is how far I have come: $\sqrt{2}\sin x = \sin x\cdot\cos x + \cos^2 x$
guest
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$x$-intercepts of secant function

I have tried setting $f(x) = 0$ and solving for $x$ by undoing the operations, and what I end up with is $x= -\pi/6$. The book gives the answer as B, however, and I haven't been able to obtain those values using symmetry. The closest I have been…
Jack
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In $\triangle ABC$, find the value of $\tan A\tan C$.

In $\triangle ABC$, line joining the circumcenter(O) and orthocenter(H) is parallel to side $AC$, then show that the value of $\tan A\tan C$ is 3. Let $\perp$ from circumcenter cuts $AC$ at D and that from orthocenter cuts it at E. Since line…
Dheeraj Kumar
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Why is $\frac{k \cdot(\sin (A) + \sin(B))}{k \cdot(\sin(A) -\sin(B))} = \frac{a+b}{a-b}$ valid in a triangle?

This is extracted from my module : In a $\displaystyle\bigtriangleup ABC$, $$\displaystyle \frac{\sin (A) + \sin(B)}{\sin(A) -\sin(B)} = \frac{k \cdot(\sin (A) + \sin(B))}{k \cdot(\sin(A) -\sin(B))} = \frac{a+b}{a-b}$$ where $A,B,C$ are the angles…
Quixotic
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