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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What is this angle in a right triangle with sides of length 5, 12, and 13?

How do I find the missing adjacent angle to leg b in a right triangle with the following side lengths: leg a = 5, leg b = 12, and hypotenuse = 13. Thanks
tay
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Sum of sin waves with same frequency and different amplitudes and phase

For the equation: $$A_1 \sin (\omega t + \theta_1) + A_2 \sin (\omega t + \theta_2) = A_3 \sin (\omega t + \theta_3)$$ I've been able to show that the amplitude of the sum is (I believe this is a standard problem): $$ A_3 = \sqrt{A_1^2 + A_2^2 + 2…
rhody
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Why is $\sin(155^\circ)$ same as $\sin(25^\circ)$?

I am practicing for a test and i've come across this question which asks "What is the value of $\sin(25^\circ)$ if $\sin(155^\circ) = 0.423$?" and I've checked on the calculator, both give same result; $0.423.$ why do they have the same value? how…
Feelsbadman
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Proving trigonometric problem from given trigonometric equations

If $p$ & $q$ are the solutions of $$a \cos x + b \sin x = c$$ Then how do I prove that, $$\cos (p + q) = \dfrac{a^2-b^2}{a^2 + b^2} $$ I tried all the adjustments I could think of, like dividing by $ \cos x $ and extracting $a$ and $b$ from the 2…
William
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Is there a formula that generalizes $\sin A+\sin B+\sin C = 4\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}$ (where $A+B+C=\pi$) to four angles?

If $A+B+C=\pi$ then we have $$\sin A+\sin B+\sin C = 4\cos\left(\frac{A}{2}\right)\cos\left(\frac{B}{2}\right)\cos\left(\frac{C}{2}\right)$$ If $A+B+C+D=\pi$ is there a similar formula?
Joshua Farrell
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Trouble proving the trigonometric identity $\frac{1-2\sin(x)}{\sec(x)}=\frac{\cos(3x)}{1+2\sin(x)}$

I have become stuck while solving a trig identity. It is: $$\frac{1-2\sin(x)}{\sec(x)}=\frac{\cos(3x)}{1+2\sin(x)}$$ I have simplified the left side as far as I…
Owen Johnstone
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Certain types of tangent values

I was messing around with values of the tangent function and came across something interesting. For example, we have $$\tan^2(\frac{\pi}{4\cdot2}) = \dfrac{\sqrt{4} - \sqrt{2}}{\sqrt{4} + \sqrt{2}}, \tan^2(\frac{\pi}{3\cdot4}) = \dfrac{\sqrt{4} -…
Cois Monis
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General form for $\sin(kx)$ in terms of $\sin(x)$ and $\cos(x)$

Identities for $\sin(2x)$ and $\sin(3x)$, as well as their cosine counterparts are very common, and can be used to synthesize identities for $\sin(4x)$ and above. Given some integer $k$, is there an equation to find $\sin(kx)$ in terms of $\sin(x)$…
user189728
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How to derive inverse hyperbolic trigonometric functions

$e^{i\theta}=\cos\theta + i\sin \theta$ $e^{i\sin^{-1}x}=\cos(\sin^{-1}x)+i\sin(\sin^{-1}x)$ $i\sin^{-1}x=\ln|\sqrt{1-x^2} + ix|$ $\sin^{-1}x=-i\ln|\sqrt{1-x^2} + ix|$ Now from here I'm kind of lost, since it seems like this should be the…
Kainui
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Solving Trigonometry Polynomial Equation.

I am trying to understand how to solve the equation $$2\sin^2x + 3\sin x + 1 = 0.$$ Please give hints.
Bilbo
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Evaluating $\big(\cot \frac{\pi}{18}-3\cot \frac{\pi}{6}\big)\cdot \big(\csc \frac{\pi}{9}+2\cot \frac{\pi}{9}\big)$

Finding value of $\displaystyle \bigg(\cot \frac{\pi}{18}-3\cot \frac{\pi}{6}\bigg)\cdot \bigg(\csc \frac{\pi}{9}+2\cot \frac{\pi}{9}\bigg)$ Try: $$\cot \frac{\pi}{18}\csc \frac{\pi}{9}-3\sqrt{3}\csc \frac{\pi}{9}+2\cot…
DXT
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$x^3 + 2x^2 + 5x + 2\cos x = 0$

$x^3 + 2x^2 + 5x + 2\cos x = 0$ How do I find the number of solutions of this equation (in $[0, 2\pi]$) without a graph? Attempt: The equation simplifies to $x(x^2 + 2x + 5)=- 2\cos x $ Minima of the quadratic occurs at $x= -1$ and it's value is…
Archer
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Why is $\cos(45°) = \frac{\sqrt{2}}{2} \simeq 0.7071$ rather than $0.5$?

I'm trying to wrap my head around trigonometry. Working in degrees we get: $$\cos(0°) = 1$$ $$\cos(90°) = 0$$ Half way between $0$ and $90$ degrees we get $45$ degrees, so it seems logical to me that $\cos(45°)$ would give $0.5$, but instead we get…
Kristoffer Helander
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Trig identity proof help

I'm trying to prove that $$ \frac{\cos(A)}{1-\tan(A)} + \frac{\sin(A)}{1-\cot(A)} = \sin(A) + \cos(A)$$ Can someone help me to get started? I've done other proofs but this one has me stumped! Just a start - I don't need the whole proof. Thanks.
PeteUK
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In a circle, parallel chords of length $2$, $3$, and $4$

I was helping my comrade answering some questions taken from review classes when I stumbled upon this question. It looks like this: In a circle, parallel chords of length $2$, $3$, and $4$ determine central angles of $A$, $B$, and $A+B$ radians,…
fitzmerl duron
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