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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Calculation of $\left(\frac{1}{\cos^2x}\right)^{\frac{1}{2}}$

Shouldn't $\left(\frac{1}{\cos^2x}\right)^{\frac{1}{2}} = |\sec(x)|$? Why does Symbolab as well as my professor (page one, also below) claim that $\left(\frac{1}{\cos^2x}\right)^{\frac{1}{2}} = \sec(x)$, which can be negative? Also, the length of a…
underdisplayname
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Solving $2\sin\left(2x\right)=3\left(1-\cos x\right)$

Background - this was part of a homework packet for students looking to skip HS pre-calc. There is a text book they use as well, but this particular problem was not in it. $$2\sin\left(2x\right)=3\left(1-\cos\left(x\right)\right)$$ My first step was…
JTP - Apologise to Monica
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Show that: $f(\theta)=\sin\theta\cos(\theta\ -k)$ is max when $\theta = \frac{k+90^{\circ}}{2}$ without using calculus.

Given $$f(\theta)=\sin\theta\cos(\theta\ -k)$$ Show that $f(\theta)$ is maximum when: $\theta = \frac{k+90^{\circ}}{2}$ I can do this easily using calculus, but I'm looking for a way of doing it without calculus. Context: A particle is projected up…
Kantura
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Simplifying $\cot\alpha(1-\cos2\alpha)$. I get $\sin 2\alpha$; book says $-4\sin\alpha$.

Please help simplify expression \begin{align}\cot\alpha\ (1-\cos2\alpha)\end{align} I tried to solve through this way: First dividing and multiplying both parts by $2$ \begin{align}\cot\alpha\ (1-\cos2\alpha)=…
Krutya
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$\cos(\alpha-\beta)+\cos(\beta-\gamma)+\cos(\gamma-\alpha)=\frac{-3}{2}$,show that $\cos\alpha+\cos\beta+\cos\gamma=\sin\alpha+\sin\beta+\sin\gamma=0$

I think that I've done a major part of the problem but I'm stuck at a point. Here's what I've done : It's given to us that $$\cos(\alpha-\beta)+\cos(\beta-\gamma)+\cos(\gamma-\alpha) = \dfrac{-3}{2}$$ Using the identity $\cos(A-B) = \cos A \cos B +…
Rajdeep Sindhu
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Deriving the expansion of $\sin (\alpha - \beta)$ using $\sin x = \sqrt{1-\cos^2 x}$

I was deriving the expansion of the expansion of $\sin (\alpha - \beta)$ given that $\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$ Now, my textbook has done it in a different manner but I thought of doing it using the…
Rajdeep Sindhu
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Solving $\sin(4k-22) = \cos(6k-13)$

My niece asked for help with an SAT prep question. We are given that $$\sin a = \cos b$$ where the angles are both acute and $a=4k-22$ and $b=6k-13$. The only way we could think to solve it is by plotting and using fzero. But since it's an SAT…
Fractal20
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If $\sec x + \csc x =p$ has four distinct solutions between $(0,2\pi)$, then which if the following is incorrect?

a) $p^2-8>0$ b) $p=\sqrt 2$ c) $p=-\sqrt 2$ d) $p=0$ My attempt $$\frac{\sec x +\csc x}{2} \ge \sqrt {\sec x \csc x}$$ $$\frac{\sin x +\cos x}{\sin x \cos x }\ge 2\sqrt {\frac{1}{\sin x \cos x}}$$ $$\sin x +\cos x \ge 2\sqrt {\sin x \cos…
Aditya
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Trouble visualizing sin and cos

I'm working on building tetris now in Java and am at the point of rotations... I originally hardcoded all of the rotations, but found that linear algebra was the better way to go. I'm trying to use a rotation matrix to rotate my pieces, and found I…
user3871
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In $\triangle PQR$, if $3\sin P+4\cos Q=6$ and $4\sin Q+3\cos P=1$, then the angle $R$ is equal to

In $\triangle PQR$, if $3\sin P+4\cos Q=6$ and $4\sin Q+3\cos P=1$, then the angle $R$ is equal to My attempt is as follows:- Squaring both equations and adding $$9+16+24\sin(P+Q)=37$$ $$\sin(P+Q)=\dfrac{1}{2}$$ either $P+Q=\dfrac{\pi}{6}$ or…
prat
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What is -cos(t) equivalent to in terms of cos(t)

I want to know if, $-\cos(t) = \cos(t+180)$ or $-\cos(t) = \cos(t-180)$ Please guide me. Thanks
user2857
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$xy<1 \iff \text{arctan }x + \text{arctan }y \in (-\pi/2,\pi/2)$

Is this claim true? $$xy<1 \iff \text{arctan }x + \text{arctan }y \in (-\pi/2,\pi/2)$$ If so, how to prove? I was led to this while trying to figure out the Addition Formula for Arctangent. I've looked at many questions and answers about that…
user693894
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Solve this equation : $\tan^{-1} \frac{x+1}{x-1} + \tan^{-1} \frac{x-1}{x} = \tan^{-1} (-7)$

Solve this equation : $\tan^{-1} \frac{x+1}{x-1} + \tan^{-1} \frac{x-1}{x} = \tan^{-1} (-7)$ This was an exam question, my try was as follows: $$ \tan^{-1} \frac{x+1}{x-1} = \tan^{-1} (-7) - \tan^{-1} \frac{x-1}{x} $$ Now, assuming that $x = \tan x…
sky-fi
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Size of the car in the picture

So I have this picture: If I print the picture, when I print this the mountain behind is around 1.3cm and the car in the lower left is around 0.4cm. I dont know how far away the car is from the mountain but I would like to know if there is a way to…
Nao
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Sine rule: Why doesn't it work in this scenario?

Why does: $$\frac{4.5}{\sin40^\circ} \not= \frac{3+3}{\sin(180^\circ - 58^\circ)}$$ Am I using the rule wrong? Any incorrect assumptions?
Wk_of_Angmar
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