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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How to Derive a Double Angle Identity.

How does one derive the following two identities: $$\begin{align*} \cos 2\theta &= 1-2\sin^2\theta\\ \sin 2\theta &= 2\sin\theta\cos\theta \end{align*}$$
Billjk
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Why are the Cosine and Sine of obtuse angles defined differently? If by convention, please explain the logic behind.

(I already know the unit circle) Why is it that the sine of an obtuse angle is the sine of its supplementary angle but the cosine of an obtuse angle is the negative of the cosine of its supplementary angle? I can see of course on the unit circle…
jeremy radcliff
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How prove $\root 4\of{\frac{1}{2}\sin x\cos z}+\root 4\of{\frac{1}{2}\cos x\sin z}=\root{12}\of{\sin 2y} $?

Let $\sin(x+y) = 2\sin\left(\dfrac{x-y}{2}\right)$ and $\sin(y+z) = 2\sin\left(\dfrac{y-z}{2}\right)$. How prove $\root 4\of{\frac{1}{2}\sin x\cos z}+\root 4\of{\frac{1}{2}\cos x\sin z}=\root{12}\of{\sin 2y} $ ?
piteer
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Rewriting $\sin(2\cos^{-1}{(x/3)})$ as a fraction?

I'm looking to simplify $$\sin(2\cos^{-1}{(x/3)})$$ I know it simplifies to $\frac{2x}{3}\sqrt{1-\frac{x^2}{9}}$, but I am unsure of the required steps. Thanks for any help
nuggetbram
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What is the relationship between the trigonometric secant and the geometric secant of a circle?

What is the difference between the geometric secant(the line that cuts two points of a curve) of a curve, and the trigonometric secant(=1/cosinex) ? If they are the same, can you explain how they are the same? Could you please explain, I am not able…
zibzabpuzzless
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Find the values of $\sin 69^{\circ},\sin 18^{\circ} , \tan 23^{\circ}$

Calculate $\sin 69^{\circ},\sin 18^{\circ} , \tan 23^{\circ}$. accurate upto two decimal places or in surds . $\begin{align}\sin 69^{\circ}&=\sin (60+9)^{\circ}\\~\\ &=\sin (60^{\circ})\cos (9^{\circ})+\cos (60^{\circ})\sin…
R K
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Finding the exact value of arctan function then adding it?

The question is $x = \arctan\frac 23 + \arctan\frac 12$. What is $\tan(x)$? I'm having trouble figuring out how to calculate the arctan values without a calculator, or do I not even need to find these values to calculate what $\tan(x)$ is? Any help…
TEEBQNE
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What is the general solution to $\sin\theta=\frac12$?

What is the general solution to $\sin\theta=\frac12$? I have an incorrect solution but I don't know why. \begin{align*} \sin\theta & =\frac12\\ \sin\theta & =\sin\alpha\\ \alpha & =\arcsin\left(\frac12\right)=\frac{\pi}{6}\\ \theta &…
CMB
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Finding $\cos(\pi/8)$ with half angle identities

I did $$\cos\left(\frac{45^\circ}{2}\right) = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$$ and ended by getting $\sqrt{\frac{2 + \sqrt{2}}{4}}$. But the answer in the book is $\frac{\sqrt{2 + \sqrt{2}}}{2}$.
Aden561
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Proving an identity, cos and sin, two variables

$$\frac{\cos(2x)+\cos(2y)}{\sin(x)+\cos(y)} = 2\cos(y)-2\sin(x)$$ The question asks to prove the identity. I tried simplifying the first half, thought maybe I could expand and simplify with the double angle formulas. Changed it to $$\cos(x)^2 -…
windy401
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Proving basic/standard trigonometric identities

How to prove the following trigonometric identities ? 1) If $\displaystyle \tan (\alpha) \cdot \tan(\beta) = 1 \text{ then } \alpha + \beta = \frac{\pi}{2}$ I tried to prove it by using the the formula for $\tan(\alpha + \beta)$ but ain't it valid…
Quixotic
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Solve $ \sqrt{2-2\cos x}+\sqrt{10-6\cos x}=\sqrt{10-6 \cos 2x} $

$$ \sqrt{2-2\cos x}+\sqrt{10-6\cos x}=\sqrt{10-6 \cos 2x} $$ I tried squaring and/or using $1-\cos x=2\sin^2{\frac{x}2}$, but no luck.
Lazar Ljubenović
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Why does $-\sec^2(x) \cot^2(x) = -\csc^2(x)$?

this is the first time I've asked a question here, so bare with me... I'm in Year 12 Maths B (kinda like Maths Extension) and, though we have not been told anything at all whatsoever about Cosecant, Secant and Cotangent, I got curious. Please excuse…
Justin
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Prove that $\sin^7 x + \cos^7 x < 1$ if $0 < x < \frac{\pi}{2}. $

I am not sure how to attack this. Using the Pythagorean identity seems just to make things messier.
Jon
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Polar Plots and square roots

When I plot a polar plot of $r=\sin (3 \theta)$, and $r=\sqrt{\sin (3 \theta)}$ I get nearly identical graphs, both $3$ pedal rose type plots. In the case without the square root, it is easy to understand the plot. However, for the plot involving…
user163862
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