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
13
votes
5 answers

Prove or disprove the implication:

Prove or disprove the implication: $a^2\cdot \tan(B-C)+ b^2\cdot \tan(C-A)+ c^2\cdot \tan(A-B)=0 \implies$ $ ABC$ is an isosceles triangle. I tried to break down the left hand side in factors, but all efforts were in vain. Does anyone have a…
medicu
  • 4,482
13
votes
2 answers

Evaluate $\cos 18^\circ$ without using the calculator

I only know $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$ as standard angles but how can I prove that $$\cos 18^\circ=\frac{1}{4}\sqrt{10+2\sqrt{5}}$$
Dr. NGILAZI BANDA
  • 659
13
votes
4 answers

Why did this extraneous root creep into the solution?

I was solving this equation and proceeded as follows: $$\arcsin (1-x) - 2\arcsin (x) = \frac{π}{2}$$ $$\implies \arcsin(1-x) = \frac{π}{2} + 2\arcsin (x)$$ $$\implies \sin (\arcsin (1-x)) = \sin \left( \frac {π}{2} + 2\arcsin (x)\right)$$ …
Myungjin Hyun
  • 309
13
votes
7 answers

The identity $\tan({\pi\over4}-{a\over2}) = \sec(a)-\tan(a)$

Today I was solving an exercise and while checking the solution on WolframAlpha, the website used the following identity: $$\tan \left({\pi\over4}-{\alpha\over2} \right) = \sec(\alpha)-\tan(\alpha)$$ Since I've never seen that formula, I tried to…
Mangusto
  • 233
13
votes
6 answers

How to expand $\cos nx$ with $\cos x$?

Multiple Angle Identities: How to expand $\cos nx$ with $\cos x$, such as $$\cos10x=512(\cos x)^{10}-1280(\cos x)^8+1120(\cos x)^6-400(\cos x)^4+50(\cos x)^2-1$$ See a list of trigonometric identities in english/ chinese
tianzhidaosunyouyu
  • 1,786
13
votes
6 answers

How do calculators evaluate inverse trig functions?

I know for simple inputs, you can just memorize the answer, but what if I wanted to find $\arcsin{0.554}$. My calculator instantly tells me that the answer is $0.5752 \ \text{radians}$. How can I do that by hand, procedurally, to always arrive at…
TechnoSam
  • 345
12
votes
2 answers

Could trigonometry exist in one dimension?

Even though trigonometry is based on circles, and angles, both of which commonly exist in two dimensions, could it also exist in one dimension? This question probably sounds really weird to you, but I was wondering in the same way that…
Kevin
  • 319
12
votes
5 answers

When simplifying $\sin(\arctan(x))$, why is negative $x$ not considered?

Let $u = \arctan(x)$, hence $x = \tan(u)$ for $u$ belongs in $(-\frac\pi2, \frac\pi2)$. Since $u$ belongs in $(-\frac\pi2, \frac\pi2)$, we consider $\sin(u)$ where $u$ belongs in $(-\frac\pi2, \frac\pi2)$. I used the unit circle to determine that…
user183826
  • 121
12
votes
3 answers

How to evaluate $\sum_{n=1}^{38}\sin\left(\frac{n^8\pi}{38}\right)$

Evaluate $$\sum_{n=1}^{38}\sin\left(\frac{n^8\pi}{38}\right)$$ I have found the problem on this page. I have no idea how to do it. Thank you very much.
kong
  • 233
12
votes
5 answers

Deriving the rest of trigonometric identities from the formulas for $\sin(A+B)$, $\sin(A-B)$, $\cos(A+B)$, and $\cos (A-B)$

I am trying to study for a test and the teacher suggest we memorize $\sin(A+B)$, $\sin(A-B)$, $\cos(A+B)$, $\cos (A-B)$, and then be able to derive the rest out of those. I have no idea how to get any of the other ones out of these, it seems almost…
Adam
  • 1,405
12
votes
3 answers

Why does $\tan(30^{\large\circ})=\frac{\tan(10^{\large\circ})\tan(50^{\large\circ})}{\tan(20^{\large\circ})}$?

This problem is based on this Facebook post. One can find the value of $x$ in this diagram by noticing that $\angle CBD=50^{\large\circ}$, and…
robjohn
  • 345,667
12
votes
1 answer

Generalisations of the identity $\tan{\frac{3\pi}{11}}+4\sin{\frac{2\pi}{11}}=\sqrt{11}$

I recently came across this curious trigonometric sum: $$\tan{\frac{3\pi}{11}}+4\sin{\frac{2\pi}{11}}=\sqrt{11}$$ which has a neat proof here: How to prove that: $\tan(3\pi/11) + 4\sin(2\pi/11) = \sqrt{11}$ For what values of $k$ does the following…
Vincent Tjeng
  • 3,304
  • 2
  • 21
  • 34
12
votes
2 answers

Are $\mathrm{arccot}(x)$ and $\arctan(1/x)$ the same function?

In my textbook it asks for me to: Prove that there is no constant $C$ such that $\text{arccot}(x) - \text{arctan}(\frac{1}{x}) = C $ for all $x \ne 0$. Explain why this does not violate the zero-derivative theorem. But I believe I have found such a…
Mark
  • 5,696
12
votes
3 answers

Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$

I'm tutoring for a college math class and we're doing putnam problems next week and this one stumped me: Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$.
user61646
  • 403
12
votes
5 answers

Solving 4 unknown angles in quadrilateral possible?

Given a quadrilateral with 4 fixed lengths, is there a way to solve for its angles? For example, I have a quadrilateral with 4 sides: 698.8m 512.5m 511.9m 695.8m How do we solve for its unknown angles?
Pacerier
  • 3,369
1 2 3
…
99 100