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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Finding $\sin(4a)$ if we know $\cos a$

I need to show that $\sin 4a = 0$ if $\cos a = 0$. I am not sure how to do this really. I know I can take $\sin^2 x + \cos^2 x = 1$ but I don't think that helps. I was also suppose to find the period of $\sin 2x + \cos 5x$, I know the answer…
Adam
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Find the measure of the smallest positive angle $\theta$ in degrees for which $\tan\theta=\frac{\cos25^\circ+\cos85^\circ}{\sin25^\circ-\sin85^\circ}$

I'm preparing for a math competition, and was stumped by this problem. The original problem is shown below, and the correct answer is $120°$. I'm posting this here to ask for explanation on how this answer was reached. The equation given is…
Nicholes Cortes
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How to solve $\frac{\sin\theta+\cos\theta}{\sec\theta+\csc\theta} = 1 /\sqrt8$

$$\frac{\sin\theta+\cos\theta}{\sec\theta+\csc\theta} = 1/ \sqrt8$$ Here is my steps $\frac{\sin\theta+\cos\theta}{\frac{\sin\theta+\cos\theta}{\sin\theta\cos\theta}} = 1/√8$ $\sin\theta+\cos\theta *…
اياد ايمن
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How to convert angles to a common orientation

I'm comparing the orientation of straight lines. I need to handle the case where the lines have the same orientation but one is drawn in the opposite direction of the other, so for example the first line's orientation is 0 and the second line's…
Hugh_Kelley
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Simplifying $\frac {\cos^4 x}{\cos^2 y}＋\frac {\sin^4 x}{\sin^2 y}＝1$

How do you simplify the following? $$\frac {\cos^4(x)}{\cos^2(y)}＋\frac {\sin^4(x)}{\sin^2(y)}＝1$$ What I've tried: $$\frac {\cos^4(x)}{\cos^2(y)}＋\frac {\sin^4(x)}{\sin^2(y)}＝1$$ $$\sin^2(x)+\cos^2(x)=1$$ $$\implies\frac…
Moustapha_M_I
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Solving the system $\cos(x)+\cos(x+y)=0$, $\cos(y)+\cos(x+y)=0$, where $0\leq x,y\leq 2\pi$

Given: $$\cos(x)+\cos(x+y)=0$$ $$\cos(y)+\cos(x+y)=0$$ $$0\leq x,y\leq 2\pi$$ find all pairs $x,y$ such that the equations hold. My attempt: Looking at the first equation $$\cos(x)+\cos(x+y)=0$$ $$\cos(x)=-\cos(x+y)$$ note…
zak zaki
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Solve for $\theta$ in $2\sec^2θ-4 =0$

Solve for $\theta$ in $2\sec^2θ-4 =0$ I have gotten toward $\sec^2θ=2$ Then $\dfrac{1}{\cos^2θ} = 2$ What is the next step to this problem?
Jon
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Find $\cos\frac{\pi}{12}$ given $\sin(\frac{\pi}{12}) = \frac{\sqrt{3} -1}{2 \sqrt{2}}$

Find $\cos\frac{\pi}{12}$ given $\sin(\frac{\pi}{12}) = \frac{\sqrt{3} -1}{2 \sqrt{2}}$ From a question I asked before this, I have trouble actually with the numbers manipulating part. Using trigo identity, $\sin^2 \frac{\pi}{12} + \cos^2…
user307640
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Why is $\sin^2 \pi/(4n)\leq 1/n^2$?

In Jeffrey Bub's Bananaworld $^1$, there's a remark (p. 101) that $$\sin^2 \frac{\pi}{4n} \;\leq\; \frac{1}{n^2} \;,$$ where $n$ is any positive integer. It's been decades since I studied trig, although I have used it occasionally with the help of…
Mars
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Calculate $\frac{4-5\sin\alpha}{2+3\cos\alpha}$

Calculate $$\dfrac{4-5\sin\alpha}{2+3\cos\alpha}$$ if $\cot\dfrac{\alpha}{2}=-\dfrac32$. My first approach was to somehow write the given expression only in terms of the given $\cot\frac{\alpha}{2}$ and just put in the value…
kormoran
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Graphing cos and transformations

I need to graph $y=-2\cos3x$ I just went the standard route and reflected across the x axis, multiplied the y axis by 2 and multiplied the x axis by three. Is this incorrect? I got the wrong answer but I am not sure why.
Adam
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Show that $4\sin^2(24^{\circ})+4\sin(24^{\circ})\sin(12^{\circ}) = 1$.

The problem asks us to show that the following equation holds true. $$4\sin^2(24^{\circ})+4\sin(24^{\circ})\sin(12^{\circ}) = 1$$ This equation has been verified on my calculator. Perhaps some basic trigonometric formulas will be enough to solve…
user1081575
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If $\sin\alpha + \cos\alpha = 0.2$, find the numerical value of $\sin2\alpha$.

If $\sin\alpha + \cos\alpha = 0.2$, find the numerical value of $\sin2\alpha$. How do I find a value for $\sin\alpha$ or $\cos\alpha$ so I can use a double angle formula? I know how to solve a problem like "If $\cos\alpha = \frac{\sqrt{3}}{2}$ ,…
mikoyan
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dot product negative angle

I have two two-dimensional unit vectors a and b. I'm trying to get their angle related to their order. arc cosine of the dot product returns the absolute value of the angle. if b is before a I want to get negative angle. e.g. if a is on 1 o'clock…
Daniel
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Convert trigonometric function to irrational fraction

for example $\cos(\frac\pi6)$ is $\frac{\sqrt3}{2}$. How can I convert any other trigonometric function into this type of fraction and preferably without a calculator?
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