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
3
votes
4 answers

Simplifying An Inverse Tan Function

I would like to know how this equality holds. $$ \tan^{-1} \frac{(2n+1) - (2n-1)}{1 + (2n+1)(2n-1)} = \tan^{-1} \frac{1}{2n-1} - \tan^{-1} \frac{1}{2n+1}.$$ I was told to use the double angle formula for $\tan \theta$ but I can't seem to show…
Jai Pancholi
  • 83
3
votes
0 answers

$2^{199}\sin(π/199)\cdots\sin(198π/199)$ Sine Product Series

What will be the value of $2^{199}\sin(\pi/199)\cdots\sin(198\pi/199)$ ? I could have found in case the functions were cosine but what should i do in case of sine?
user220382
3
votes
2 answers

Finding value of $\sin (15\, ^{\circ})$ with half angle identity

The answer I got when trying to solve it was $\sqrt{\frac{1 - \sqrt3}{2} }$ but the book says it's $\sqrt{ \frac{2 - \sqrt3}{2}}$ and I don't know how the two on the top half gets there.
windy401
  • 1,211
3
votes
2 answers

Identifying functions equivalent to $y=-3\sin x+2$

I was given the following problem on a quiz: I put A, C, and D. The answer was A and D. We were taught four relevant equations: $\sin(x)=-\sin(-x)$ $\cos(x)=\cos(-x)$ $\sin(x)=\cos(x-\frac{\pi}{2})$ $\cos(x)=\sin(x+\frac{\pi}{2})$ Based on my…
mowwwalker
  • 1,201
3
votes
5 answers

Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $

Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $ Can someone please explain to me how to go about doing this?
user860374
  • 4,131
3
votes
3 answers

Solve $\left| \cos { \left( 2x \right) } \right| = \frac { 1 }{ 2 } $

How do you solve the following equation over an unrestricted domain; $$\left| \cos { \left( 2x \right) } \right| = \frac { 1 }{ 2 } $$ I can solve half of it; $$\cos { \left( 2x \right) } =\frac { 1 }{ 2 } \\ x=\pi n\pm \frac { \pi }{ 6 } $$ but…
user222974
  • 31
3
votes
2 answers

Prove the identity $\sin^4α-\cos^4α=2\sin^2α-1$

Prove the identity $\sin^4α-\cos^4α=2\sin^2α-1$ Well, I thought to start it this way: $$(\sin^2α-\cos^2α)(\sin^2α+\cos^2α)=2\sin^2α-1=>\\(\sin α-\cos α)(\sin α+\cos α)(\sin^2α+\cos^2α)=2\sin^2α-1$$ I don't know how to continue...
seda
  • 125
3
votes
3 answers

Is it generally true that $\arcsin \theta + \arccos \theta = \frac{\pi}{2}$?

Verify that: $\arcsin \theta+\arccos \theta=\frac{\pi}{2}.$ (1) How can one verify (1) when it is not generally true? We can rewrite (1) as Verify that if $\sin u = \cos v$, then $u+v=\frac{\pi}{2}$. What about $\sin \frac{2\pi}{3}$ and $\cos…
Samama Fahim
  • 1,459
  • 4
  • 24
  • 39
3
votes
1 answer

I need easy solutions to these trigonometric equations

I need easy solutions to these trigonometric equations: $$\sin^3x \cos x = \frac{1}{4} \text{ and }\sin^4x \cos x = \frac{1}{4}$$
Adam
  • 835
  • 2
  • 8
  • 14
3
votes
8 answers

Why is the tangent of 22.5 degrees not 1/2?

Sorry for the stupid question, but why is the tangent of 22.5 degrees not 1/2? (Okay... I get that that the tangent of 45 degrees is 1 ("opposite" =1, "adjacent" =1, 1/1 = 1. Cool. I am good with that.) Along those same lines, if the "opposite"…
Jim
  • 31
3
votes
3 answers

Is it true that $a\cos \alpha \theta = b \cos \beta \theta \implies a=b$ and $\alpha = \beta$

Is it true that $\forall \theta:a\cos \alpha \theta = b \cos \beta \theta \implies a=b$ and $\alpha = \beta$? If so, how do I prove it? I know it isn't true for the sine case since we could have $a=-b$ and $\alpha = - \beta$
user85798
  • 1
3
votes
4 answers

Can $f(x)=\sin(x) + \cos^2(x)$ take the value $\sqrt{2}$?

Can $f(x)$ take the value of $\sqrt{2}$, where $$f(x) = \sin(x) + (\cos(x))^2\quad ?$$ When equating the value of $f(x)$ to $\sqrt{2}$, it gives imaginary values of $\sin(x)$. Thanks in advance.
vikiiii
  • 2,681
  • 2
  • 34
  • 45
3
votes
2 answers

Trig Identity / Pythagorean Theorem confusion?

I run into a problem when I'm trying to prove how $\tan^2x+1 = \sec^2x$, and $1+\cot^2x=\csc^2x$ I understand that $\sin^2x+\cos^2x = 1$. (To my understanding 1 is the Hypotenuse, please correct me if I'm wrong). If referring to a Pythagorean…
YangCPG
  • 71
3
votes
3 answers

proving trigonometric identity involving 2 arguments

$\cos^2\theta=\dfrac{m^2-1}{3}$, $\tan^3\dfrac{\theta}{2}=\tan\alpha$. How to prove $$\cos^\frac{2}{3}\alpha+\sin^\frac{2}{3}\alpha=\frac{2}{m}^{2/3}. $$ I got the following $$\tan^\frac{2}{3}\alpha=\frac{1- \sqrt\frac{m^2 -…
rajiv
  • 31
3
votes
4 answers

How to calculate sin(65) without a calculator.

I know about the sum and difference formula but I can't think of two values which will be able to use for sin(65). Therefore, I come to the question: How to calculate sin(65) without a calculator.
Justin
  • 1,225
1 2 3
…
99 100