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.

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How do I prove this seemingly simple trigonometric identity

$$a = \sin\theta+\sin\phi\\b=\tan\theta+\tan\phi\\c=\sec\theta+\sec\phi$$ Show that, $8bc=a[4b^2 + (b^2-c^2)^2]$ I tried to solve this for hours and have gotten no-where. Here's what I've got so far : $$ \\a=…
user126456
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Solving sin z = -z in reals

How do we prove that z=0 is the only real solution? I tried examining cases and quarters, but not sure how it is rigorously proven.
Jackie Poehler
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how to find mid point of an arc?

I have start point $(x_1,y_1)$ and an end point $(x_2,y_2)$ and radius of arc. How to calculate the co-ordinates of mid-poing of arc? The arc is the part of a circle. Known Values length of AD // that is radius B(x,y) C(x,y) Needs to find D(x,y) …
coure2011
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Is my trig result unique?

I recently determined that for all integers $a$ and $b$ such that $a\neq b$ and $b\neq 0$, $$ \arctan\left(\frac{a}{b}\right) + \frac{\pi}{4} = \arctan\left(\frac{b+a}{b-a}\right) $$ This implies that 45 degrees away from any angle with a rational…
Brian Abend
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Identity for a weighted sum of sines / sines with different amplitudes

I'm trying to simplify the following sum of sines with different amplitudes $$ a \sin(\theta) + b \sin(\phi) = ??? \,\,\,\,\, (1) $$ I know that $$ a \sin(\theta) + a \sin(\phi) =…
daniloz
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relationship of polar unit vectors to rectangular

I'm looking at page 16 of Fleisch's Student's Guide to Vectors and Tensors. The author is talking about the relationship between the unit vector in 2D rectangular vs polar coordinate systems. They give these equations: \begin{align}\hat{r} &=…
Peter
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calculator's angle answer for trig ratios that can work in more than 1 quadrant on the unit circle

Why does the calculator do a cc (counterclockwise) rotation for positive trig ratios instead of clockwise, and a clockwise rotation for negative sine & tan instead of cc and a counterclockwise rotation for negative cos ratios instead of a…
Allan Henriques
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Unsure if I have solved/proven this trigonometric inequality.

This is my first post here. I apologize if it goes against any guidelines for posting. I study math as a hobby and am currently dealing with trigonometry on a high school level. I have so far learned the formulas for trigonometric addition and…
Grenadine
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$x_{0}= \cos \frac{2\pi }{21}+ \cos \frac{8\pi }{21}+ \cos\frac{10\pi }{21}$

Prove that $x_{0}= \cos \frac{2\pi }{21}+ \cos \frac{8\pi }{21}+ \cos\frac{10\pi }{21}$ is a solution of the equation $$4x^{3}+ 2x^{2}- 7x- 5= 0$$ My try: If $x_{0}$, $x_{1}$, $x_{2}$ be the solutions of the equation…
user530708
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When is $\sin(x) = \cos(x)$?

How do I solve the following equation? $$\cos(x) - \sin(x) = 0$$ I need to find the minimum and maximum of this function: $$f(x) = \frac{\tan(x)}{(1+\tan(x)^2}$$ I differentiated it, and in order to find the stationary points I need to put the…
NPLS
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How to see $\sin x + \cos x$

$$\sin x + \cos x = \sqrt{2} \sin(x + \pi/4)$$ Is there an easy way to visualize this identity or to convert the left-hand side to the right-hand side? In general, can $p \sin x + q \cos x$ for some integers $p$ and $q$ be easily expressed as…
hollow7
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tricky system of trigonometric equations

I am not very fresh in math, but I need to solve this system: \begin{gather} A\sin(x-y)+B\sin(z-y)=C\\ A\cos(x-y)+B\cos(z-y)=D \end{gather} where $A,B,C,D$ and $x$ are given. I tried to expand and combine the bracket terms and I suppose that there…
valbe
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What is the correct way to solve $\sin(2x)=\sin(x)$

I've found two different ways to solve this trigonometric equation $\begin{align*} \sin(2x)=\sin(x) \Leftrightarrow \\\\ 2\sin(x)\cos(x)=\sin(x)\Leftrightarrow \\\\ 2\sin(x)\cos(x)-\sin(x)=0 \Leftrightarrow\\\\ \sin(x) \left[2\cos(x)-1 \right]=0…
user24047
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4 answers

$\sec\theta+\tan\theta=p$ and $\sec\theta\tan\theta=q$. Eliminate $\theta$ to form a equation between $p$ and $q$.

$\sec\theta+\tan\theta=p$ and $\sec\theta\tan\theta=q$. Eliminate $\theta$ to form a equation between $p$ and…
diya
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When does this equation $\cos(\alpha + \beta) = \cos(\alpha) + \cos(\beta)$ hold?

I come across this problem in an advanced maths textbook for grade 11 in my country. And it's marked a star, which means that it's a difficult exercise, and so, no solution for this problem is given. I can solve problems asking for which conditions…
user49685
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