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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Why did not I find all the solutions to this equation? $\sin{3x}+\cos{2x}-\sin{x}=1$

$x$ needs to be in $[0,\pi]$ What I…
Vinícius
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Why is the cosine of a right angle, 90 degrees, equal to zero?

Why the cosine of an angle of 90 degree is equal to zero? By definition we know that: $$\text{cos } \alpha = \frac{\text{adjacent}}{\text{hypotenuse}}.$$ If we want to apply the definition to the situation in the image below: we have…
JB-Franco
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Finding $ \csc \theta $ given $ \cot \theta $

I have the following problem: If $ \cot{C} = \frac{\sqrt{3}}{7} $, find $ \csc{C} $ From my trig identities, I know that $ \cot{\theta} = \frac{1}{\tan{\theta}} $, and $ \csc{\theta} = \frac{1}{\sin{\theta}} $, and also $ \cot{\theta} =…
friedo
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Proof of the radical expression of $\cos\dfrac {2\pi}{17}$

Question: How would you prove the equation$$\small\cos\dfrac {2\pi}{17}=\dfrac {-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}}{16}\tag1$$ I'm not too sure how to prove it and I'm not sure where…
Frank
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Solve $2\cdot \sin x \cdot \sin (50°+2x)=\sin (50°)$

I am trying to find a solution for the problem: $$2\cdot\sin x \cdot \sin (50°+2x)=\sin (50°), \quad x\in \left[0,\frac{\pi}{2}\right]$$ Approach: I can check, by inspection, that $x=50°$ is a solution. I have tried open $\sin(50º+2x)$ but it didn't…
Arnaldo
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how to calculate the exact value of $\tan \frac{\pi}{10}$

I have an extra homework: to calculate the exact value of $ \tan \frac{\pi}{10}$. From WolframAlpha calculator I know that it's $\sqrt{1-\frac{2}{\sqrt{5}}} $, but i have no idea how to calculate that. Thank you in advance, Greg
Greg Witczak
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On the proof $\tan 70°-\tan 20° -2 \tan 40°=4\tan 10°$

I am currently studying in class 10 and I am unable to do this problem. $$\tan 70 ° -\tan 20° -2 \tan 40° =4\tan 10°$$ Can anybody please help me. Thanks!
user354443
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In a $\triangle ABC,a^2+b^2+c^2=ac+ab\sqrt3$,then the triangle is

In a $\triangle ABC,a^2+b^2+c^2=ac+ab\sqrt3$,then the triangle is $(A)$equilateral $(B)$isosceles $(C)$right angled $(D)$none of these The given condition is $a^2+b^2+c^2=ac+ab\sqrt3$. Using sine rule, $a=2R\sin A,b=2R\sin B,c=2R\sin C$,we…
Vinod Kumar Punia
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Proving:$\tan(20^{\circ})\cdot \tan(30^{\circ}) \cdot \tan(40^{\circ})=\tan(10^{\circ})$

how to prove that : $\tan20^{\circ}.\tan30^{\circ}.\tan40^{\circ}=\tan10^{\circ}$? I know how to prove $ \frac{\tan 20^{0}\cdot\tan 30^{0}}{\tan 10^{0}}=\tan 50^{0}, $ in this way: $ \tan{20^0} =…
Frank
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Expressing $\sin(2x)-8\cos(2x)$ as a single sine function

I am asked as a part of a question to express $\sin(2x)-8\cos(2x)$ as a single sine function. I know it has something to do with the trigonometric identity $$\sin(a-b)=\sin(a) \cos(b)-\cos(a)\sin(b)$$ but I can't get my head around it because of…
εν οίδα ότι ουδέν οίδα
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Is $\sqrt{\cos^2(x)} = |\cos x|$?

I am taking the principal root of $\cos^2(x)$ so I thought it would be, but when you ask wolfram alpha it says it's only sometimes true, when $x > 0$ (see here). Can someone explain why this is to me and give a value of $x$ for which it is not true?…
John
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Solve $\cos{(7x)}+\sin{(3x)}=0$

Solve $\cos(7x)+\sin(3x)=0$ I'm stuck. Help me, please. I did $\cos(7x)=\cos(4x+3x)=\cos(4x)\cos(3x)-\sin(4x)\sin(3x)$ So the original equation becomes $\cos(4x)\cos(3x)-\sin(4x)\sin(3x)+\sin(3x)=0$ But that became long and ugly. What can I do now?
user276290
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Help with calculating the angle to turn towards a target in a coordinatesystem

I know the following: my own position my own facing (the angle im turned) my targets position What i would like some help with is how i calculate the shortes way to turn and the angle to turn. If i my position was (4,4) and i faced 90 degrees and…
Jason94
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Find the sum: $\sum_{i=1}^{n}\dfrac{1}{4^i\cdot\cos^2\dfrac{a}{2^i}}$

Find the sum of the following : $S=\dfrac{1}{4\cos^2\dfrac{a}{2}}+\dfrac{1}{4^2\cos^2\dfrac{a}{2^2}}+...+\dfrac{1}{4^n\cos^2\dfrac{a}{2^n}}$
lzutao
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What is the most elegant and simple proof for the law of cosines?

Given 2 sides, and an angle between those two sides, what is the simplest proof you can come up with to find the measure of the 3rd side?
user8210
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