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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If $t=\tan{\frac{x}{2}}$, then $\cos{x}$ can be expressed as...

If $t=\tan{\frac{x}{2}}$, then $\cos{x}$ can be expressed as a) $\frac{1+t^2}{1-t^2}$ b) $\frac{2t}{1+t^2}$ c) $\frac{1-t^2}{1+t^2}$ d) $\frac{2t}{1-t^2}$ Attempt: I tried using the half angle formula but it just leaves me with an expression in…
Parseval
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Proving the trig identity $\sin(20^\circ)\cos(65^\circ)-\cos( 20^\circ)\sin(65^\circ)$

Using the trigonometric identities I have to prove that: $$ \sin(20^\circ)\cos(65^\circ)-\cos(20^\circ)\sin(65^\circ)=-\frac{1}{\sqrt{2}} $$ I solved $$ \sin(20^\circ)\cos(65^\circ) = \frac{\sin(-45^\circ)+\sin(85^\circ)}{2}$$ and…
super95
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Tangent half angle formula

So we start with the following tangent half angle formula: $$ \tan\left(\frac \theta2\right) = \pm\sqrt{\frac {1 - \cos \theta}{1 + \cos \theta}} $$ If I do some algebraic manipulation I end up with the following below: $$ \tan \left(\frac…
Omicron
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Methods for finding $\cos(4x)$ given that $\sin(2x) = \frac{3}{5}$

If $\sin(2x) = \frac{3}{5}$ Find $\cos(4x)$.. I tried by : $\cos(4x)= \cos(2\cdot2x)$.. And $\cos (2\cdot2x) = 1-2\sin^2(2x)$ .. From it ---- $\cos(4x)=0.28$. Is there any other ways ?
Hamzeh K. Yacoub
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Values for $a$ and $b$ in $y=\cos(x)+a\cos(bx)$ such that every real value for $x$ has either a positive or $0$ value for $y$

If there is a function in the form $y=\cos(x)+a\cos(bx)$ do there exists real number values for $a$ and $b$ such that for every real number value for $x$ there is either a positive number value for $y$ or a $0$ value for $y$?
Anders Gustafson
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Trigonometric equation $\cos (5x) = \sin (x)$ - How to find $5x$?

I want to find the general solution of the following equation for $x \in \mathbb{R}$: $$\cos (5x) = \sin (x)$$ I know it might sound silly, but I don't know how to bring $5x$.
Rat Rod
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proving $\cos (A+B)>0$, if given angles $A$ and $B$

If $\displaystyle A=3\sin^{-1}\left(\frac{6}{11}\right)$ and $\displaystyle B = 3\cos^{-1}\left(\frac{4}{9}\right),$ then proving $\cos (A+B)>0$ Attempt: $$…
DXT
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Prove that $\sin\frac{\pi}{7}\sin\frac{2\pi}{7}\sin\frac{3\pi}{7}=\frac{\sqrt{7}}{8}$.

Prove that $\sin\frac{\pi}{7}\sin\frac{2\pi}{7}\sin\frac{3\pi}{7}=\frac{\sqrt{7}}{8}$. What I've tried doing : If $\theta=\frac{\pi}{7}:$ $$ 3\theta+4\theta=\pi $$ This allowed me to prove that…
Vishnu V.S
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Prove that $\cos 20^{\circ} + \cos 100^{\circ} + \cos {140^{\circ}} = 0$

Assume $A = \cos 20^{\circ} + \cos 100^{\circ} + \cos 140^{\circ}$ . Prove that value of $A$ is zero. My try : $A = 2\cos 60^{\circ} \cos 40^{\circ} + \cos 140^{\circ}$ and I'm stuck here
S.H.W
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Find the value of $\arctan(1/3)$

How can I calculate $\arctan\left({1\over 3}\right)$ in terms of $\pi$ ? I know that $\tan^2(\frac{\pi}{6})= {1\over3}$ but don't know if that helps in any way.
Liviu
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How to find out coefficient $b_1$ in summation $\sin n\theta=\sum_{r=0}^{n} b_r \sin^r\theta$

Let $n$ be a odd integer. If $$\sin n\theta=\sum_{r=0}^{n} b_r \sin^r\theta$$ for every value of $\theta$. I have to find $b_1$. I don't know how to start this? Thanks.
J. Deff
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Various methods to find value of $\sin 18^\circ$

To find value of $\sin 18^\circ$. Now my textbook gives a proof in which it takes $\theta=18^\circ$ and then multiply it out by 5 and write again as sum of $2\theta+3\theta$ and then taking sin on both sides forms a quadratic equation. Are there any…
Gathdi
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To prove $\frac{1}{\sin 10^\circ}-\frac{\sqrt 3}{\cos 10^\circ}=4$

To prove: $$\frac{1}{\sin 10^\circ}-\frac{\sqrt 3}{\cos 10^\circ}=4$$ I tried taking lcm but could not get to anything.
Sophie Clad
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Writing sins 3s in terms of sin s

In the reduction below, I do not understand line 4 and 6. What identities were applied to line 3 and 5 to reach those conclusions? How were those identities introduced? 1 ) $\sin 3a $ 2 ) $= \sin(2a +s)$ 3 ) $= \sin2a ·\cos a + \cos 2a·\sin…
Ciambro
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Prove that: $\cos^2 20° + \cos^2 40° +\cos^2 80° = \sin^2 20° + \sin^2 40° + \sin^2 80°$

Prove that: $\cos^2 20° + \cos^2 40° +\cos^2 80° = \sin^2 20° + \sin^2 40° + \sin^2 80°$ My Attempt: $$L.H.S=\cos^2 20° + \cos^2 40° + \cos^2 80°$$ $$=\dfrac {1+\cos 40}{2}+\dfrac {1+\cos 80}{2} + \dfrac {1+\cos 160°}{2}$$ $$=\dfrac {3+\cos 40°+\cos…
pi-π
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