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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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$\tan( x + i y ) = a + ib$ then $\tan (x - iy) = a - ib $?

How to prove, if $\tan( x + i y ) = a + ib$ then $\tan (x - iy) = a - ib $ ? I am not familiar with trignometric identities. So any help will be appreciated. Thanks in Advance.
GA316
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$\arctan(-3/2)$ doesn't give expected result.

Let's say I want to find the angle measure (in degrees) such that $\tan(x) = -3/2$. It turns out that $x \approx 123.7$, and when I compute $\tan(123.7)$, I get $\approx -3/2$; so far so good. However, when I compute $\arctan(-3/2)$, I get $\approx…
jeremy radcliff
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If $0\leq a \leq 3; 0\leq b \leq 3$ and the equation $x^2 +4+3 cos(ax+b)=2x$ has at least one solution , then find the value of a+b

Problem : If $0\leq a \leq 3; 0\leq b \leq 3$ and the equation $x^2 +4+3 cos(ax+b)=2x$ has at least one solution , then find the value of a+b. Solution : We can write the given equation : $x^2 +4+3 cos(ax+b)=2x$ as $x^2-2x+4 =-3cos(ax+b)$ Since…
user108258
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Proving that $\tan(x) = x$ has exactly one solution per interval $((n-\frac12)\pi, (n+\frac12)\pi)$

I want to prove that $\tan(x) = x$ has exactly one solution per interval $((n-\frac12)\pi, (n+\frac12))$. My attempt: $\tan(x)$ is $\pi$-harmonic, and has a range of $(-\infty, \infty)$ for each interval $(\frac\pi2n, \frac\pi2(n+1)$), and is…
Alec
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Basic Trigonometry Question

If $\cos{(A-B)}=\frac{3}{5}$ and $\sin{(A+B)}=\frac{12}{13}$, then find $\cos{(2B)}$. Correct answer = 63/65. I tried all identities I know but I have no idea how to proceed.
Kanishk
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Understanding trig interval

I have kind of a random question I'm hoping someone could help me with. So I was thinking about the interval $[-\pi, \pi]$ for a trig functions. Isn't this is the same interval as $[0, 2\pi]?$ The reason why I say that (and maybe this is where my…
Astro
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How is $-(\pi - \tan^{-1}(1/\sqrt{3}))=-5 \pi/6$

How did they get from 2nd last to last step? $$-\Bigg(\pi - \tan^{-1}\bigg({\frac{1}{\sqrt{3}}}\bigg)\Bigg)=-\frac{5 \pi}{6}$$
Jiew Meng
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Practical trigonometry question I can't figure out (Highschool Level)

"Jack is on a bearing of 260 degrees from Jill. What is Jill's bearing from Jack?" The answer is 080 degrees. I really can't figure out how. Any help is appreciated.
PrincipiaMathematica
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How does $\tan50^\circ$ compare to $1$?

What is $\tan50^\circ$? (without using a calculator) 1 a little less than 1 a little bigger than 1 none of the above answers I think the answer is 3, but I can not explain this mathematically. The only logic I came up with is that…
user155910
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Why is $\sin(\pi/6) = 0.5$?

Why is $\sin(30^\circ)$ exactly $0.5,$ when it could be 0.49999 or something else? There must be an easy geometric explanation?
Rodrigo
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How to obtain the exact value of $\sin(3n)$?

Suppose $3n$ is an angle between $0$ and $90$ degrees and that $n$ is an integer. By considering half squares and half equilateral triangles it's easy to obtain expressions for $\sin(3n)$ if $n=10$, $n=15$ and $n=20$. (By "expressions" I mean closed…
poolpt
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Trigonometry equation solution

I have to solve this equation for all solutions $$\sin(2x) = -\cos(2x)$$ Here are my steps $$\sin(2x) + \cos(2x) = 0$$ $$\cos(2x)(\tan(2x) + 1) = 0$$ Upon solving these two equations, I find: For cosine, $x = \frac{\pi}{4} + \pi n$ For both, $x =…
Jason
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A little Problem in Trigonometry (Multiple Angle)

If $\tan^2 \theta = 1 + 2\tan^2 \phi$, show that $\cos 2\phi = 1 + 2\cos2\theta$. What I have done.. $$\implies \tan^2 \theta = 1 + 2\tan^2 \phi\\ \implies 1 + \tan^2 \theta = 2 + 2\tan^2 \phi\\ \implies 1 + \tan^2 \theta = 2(1 + \tan^2…
Swetank
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Trigonometric Question: $\sqrt2\sin10 (\sec5+\frac{2\cos 40}{\sin5}-4\sin35)=...$

How to compute the following trigonometric question $$\sqrt2\sin10 (\sec5+\frac{2\cos 40}{\sin5}-4\sin35)=...$$ I am having problem to solve this trigonometric question. I tried to use identity $\sin10=2\sin5\cos5,\cos40=\cos(45-5), and…
Venus
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Trigonometry Problem. Help me!

Simplify $$\frac{\cos^{2}a-\cot^{2}a +1}{\sin^{2}a + \tan^{2} a -1}$$ Please help me solve this problem
Jung Hyun Ran
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