Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.

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Understanding trigonometric identities

Can someone help me understand trigonometric identities? For example, it is known that $\cos(90-\theta)$ is equal to $\sin \theta$, and vice versa. But why? Is it something to do with the unit circle? ...
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2answers
74 views

Rectangular to polar form using exact values.

I'm in a first year math course at university, and we've been asked to convert a rectangular form complex number into polar form, using exact values only. I have the modulus, that's all good. But I ...
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6answers
176 views

How do I solve $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{4\,dx}{\sin^2(x)\cos^2(x)}$?

Alright so I have $$\int_{\pi/6}^{\pi/4}\frac{4\,dx}{\sin^2(x)\cos^2(x)}.$$ And I am not completely sure on how to tackle this problem. All I have done thus far is ...
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2answers
87 views

How do I find a missing angle using a reciprocal trigonometric function?

I just attempted this as best as I could, but I'm not sure if I'm correct. Here's the work: $$\cot x =\frac{1}{2}$$ $$\frac{1}{\tan{x}} = \frac{1}{\frac{1}{2}}$$ $$\frac{1}{\tan^{-1}\cdot\tan x} = ...
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1answer
38 views

Solve trigonometric function $x_1 \sin(2\alpha)+x_2 \cos(2\alpha) - x_3 \sin(\alpha) - x_4 \cos(\alpha) = 0$

I need to solve a trigonometric function similar to the following one for $\alpha$. $$ x_1 \sin(2\alpha)+x_2 \cos(2\alpha) - x_3 \sin(\alpha) - x_4 \cos(\alpha) = 0 $$ I found a solution to a very ...
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2answers
50 views

finding the value of $Z+Z^{2}+Z^{3}… $ if…

If $ Z+Z^{-1} = 2 \cos 5$ then what's the value of $Z+Z^{2}+Z^{3}.... ......Z^{63}$. I wanted to to solve this with the value of $Z$. But may be the value of $Z$ is complex. Now it's quite impossible ...
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6answers
120 views

How to prove $\tan^{-1}(n+1)-\tan^{-1}(n-1)=\tan^{-1}\big(\frac{2}{n^2}\big)$?

Prove $$\tan^{-1}(n+1)-\tan^{-1}(n-1)=\tan^{-1}\big(\frac{2}{n^2}\big)$$ for $n \ge 1$ If I use mathematical induction how do I manipulate the numbers to fit in the induction hypothesis? Is there ...
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4answers
897 views

What is the most efficient way to calculate the sine of a rational number?

I'm happy that we can use some trig identities like $$\sin\left(\frac{\theta}{2}\right) \equiv \pm \sqrt{\frac{1-\cos(\theta)}{2}}$$ and $$\sin(\alpha \pm\beta) \equiv \sin(\alpha) \cos(\beta)\pm ...
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3answers
49 views

Trigonometric Identities involving fractions

The question is to simplify: However, when I do that I end up with: $\frac{\cos\theta}{\frac{1}{\cos\theta}}$ Now, I don't know how to deal with these types of fractions. I have not encountered ...
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204 views

Given the sample triangle below and the conditions, find the hypotenuse of the triangle

Given the sample triangle below and the conditions $b=\frac53$ and $a=16$, find the hypotenuse of the triangle.
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500 views

Trigonometric simplification for limits.

Have to evaluate this limit, but trigonometry part is :( $$\lim_{x\to 0} \dfrac{1-\cos^3 x}{x\sin2x}.$$ Had written the denominator as $2x\sin x\cos x$, no idea what to do next. Please help...
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4answers
328 views

What does "sin$\theta > 0$ mean here?

The question is: If $\tan$ $\theta$ = -$\frac{8}{15}$, and $\sin$ $\theta$ > $0$, find $\cos$ $\theta$. What I did was draw a triangle on the unit circle with sides 8, 15 and therefore ...
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2answers
56 views

Exact value of a trigonometric ratio

I was asked to find the exact value of $\tan 240^\circ$. On my calculator, I type $\tan 240^\circ$, and then square the value to get a final answer of $\sqrt3$. However, the textbook answer says the ...
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1answer
101 views

Proof involving an isosceles triangle

I came across this problem in some (maybe) high school book: Let $ABC$ be an isosceles triangle s.t. $AB=AC$. Also, $\alpha>\beta$. It is known/given: ...
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4answers
164 views

About $\sin 2\theta+\sqrt{3}\cos 2\theta=-\frac{\sqrt{3}}{2}$.

$0\leq\theta<2\pi$. When $\theta$ satisfies $\sin 2\theta+\sqrt{3}\cos 2\theta=-\frac{\sqrt{3}}{2}$, solve $\alpha+\beta$ ( $\alpha$:= minimum $\theta$, $\beta$:= maximum $\theta$). From the graph ...
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38 views

Trigonometric Proof:

Question: If $m\cos\alpha-n\sin\alpha=p$ then prove that $m\sin\alpha+n\cos\alpha=\pm \sqrt{m^2+n^2-p^2}$ My Efforts: $(m\cos\alpha-n\sin\alpha)^{2}=p^2$ ...
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2answers
109 views

Simplifying difference trig expression

Rewrite the following expression as a simplified expression containing one term: $$\cos (\frac{\pi}{3}+\varphi) \cos (\frac{\pi}{3}-\varphi) - \sin (\frac{\pi}{3}+\varphi) \sin ...
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1answer
253 views

How to solve the trigonometric equation $\cos (\pi\theta/\beta) - \cos(2\pi\theta/\beta)=0$?

I have a question regarding a problem I've been attempting to solve. It is an acceleration equation: $$a = ...
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1answer
327 views

Finding exact values of trig functions

Find exact value of each trigonometric function of $\theta$ if $\tan\theta=-1/5$ and $\sec \theta >0$ I know that $\cot \theta=-5,$ right? Secant and cosine are positive in the fourth ...
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4answers
67 views

Rewriting trigonometric expression in terms of $\cot x$

Rewrite the following expression in terms of $\cot x$: $$\frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x}$$ I usually show my work on this site but I'm really lost about this problem. Any help ...
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2answers
857 views

Using sketch to find exact value of trigonometric expression

Use sketch to find exact value of $\tan (\cos^{-1}\dfrac{5}{13})$ I drew a right triangle with angle $\theta$ and sides $12,5,3.$ If $\cos \theta=\frac{5}{13},$ then $\sin \theta = \frac{12}{13}$ ...
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3answers
51 views

Technique to find the number of solutions…

The question is : find the number of solutions of $|\sin(x)| = |\cos3x|$ in $[-2\pi , 2\pi]$ . I ve seen the graph in Wolfram alpha graph plotter and found 24 solutions . But I want to know ...
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3answers
790 views

Trigonometric proof [L.H.S.=R.H.S]

Question: $$\frac{2-3\sin\theta+\sin^3\theta}{\sin\theta+2}=2\sin\theta (\sin\theta-1)+\cos^2\theta$$ I don't know how to start with these problem. Normally these type of proof confuse me. In my ...
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0answers
25 views

Proof that function is real part of $\sec(z)$ [duplicate]

I'm working on the following problem: I've deduced that the key is to show that $u$ is the real part of $\sec(z)$. But, I'm getting stuck in the algebra and am hoping someone can point me in the ...
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1answer
59 views

Proof that $\cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)$

Could anyone offer a proof that $$ \cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)? $$
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2answers
45 views

Why is $f(x)=\sin (2x-\pi)$ the same as $g(x)=-\sin(2x)$?

Why is $f(x)=\sin (2x-\pi)$ the same as $g(x)=-\sin(2x)$. Wolfram Alpha gives me this result? Where is the mistake in my reasoning?: Step A: I start with $f(x)=\sin(x)$ It's period is $2\pi$. Step ...
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3answers
71 views

Why was the inverse sine ratio used? and not sine?

From what point should I look when determining what trig ratio to use? If they can use hypotenuse over opposite, they can also use opposite over hypotenuse. Were should I look to determine the ...
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1answer
279 views

Splitting a triangle to make two equal halves, find the length of the new line

Could someone please explain to me how I would find this out? I have a triangle and I need to find the length of the line that would split it down the middle so that the areas were even. A = 105 ...
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1answer
200 views

Polar form of complex numbers.

Write the given number in polar form $re^{i\theta}$ i) $z = -8\pi (1 + \sqrt{3}i)$ So I thought that $\theta = \arctan(-\sqrt{3}/-1) = \frac{4\pi}{3}$ and it would be $z = 8\pi ...
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2answers
166 views

Find the length of a diagonal of a city block.

A city block is a square with each side measuring 104 yards. Find the length of the diagonal of a city block.
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42 views

$\sin x+\sqrt{5} \cos x$ in a form $c\cdot \sin (x+d)$

How can I rewrite $\sin x + \sqrt{5}\cdot \cos x$ in a form $c \cdot \sin (x+d)$??? How can I find the values for $c$ and $d$? I have no idea how to solve that algebraically. Is there also a ...
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3answers
62 views

Trigonometric identity, simplifying an expression to $(1-\sin^2 a\cos^2a)/(2+\sin^2a\cos^2a)$

Question: $$\left(\frac{1}{\sec^2A-\cos^2A}+\frac{1}{\csc^2A -\sin^2A}\right)\sin^2A\cos^2A=\frac{1-\sin^2A \cos^2A}{2+\sin^2A\ \cos^2A}$$ Prove L.H.S. = R.H.S. My Efforts: ...
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1answer
43 views

Question about writing cyclometric function in function of x

I have an excercise about cyclometric functions and I'm stuck right now: $\cot(2*arcsec(x))$ Let $\mathbb y=arc\ sec(x) \Leftrightarrow sec(y)=x$ then $cot(2y)=\frac {cos(2y)}{sin(2y)}=\frac ...
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1answer
115 views

Can Angles A and B In A Trapezium Be Solved Using Basic Geometry?

Can angles A and B be solved? Neither the area nor the perimeter was given. Thank you very much if you can help! :)
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2answers
49 views

An arctan problem including a diophantine equation

This is a follow-up question to An equation of the form A + B + C = ABC . I totally messed up with making the equation from the question specification . Actually the question was $$ ...
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2answers
144 views

Angle and circle intersection, find the circular segment area

Playing Kerbal Space Program, I found myself wondering about what a satellite would see of a planet depending on its field of view and its altitude. I tried attacking the problem from various angles ...
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1answer
84 views

Help needed in verifying a trigonometric identity

I have the following identity: $$32\sin^{2}\left(\theta\right)\cos^{4}\left(\theta\right) =2 + \cos\left(2\theta\right) - 2\cos\left(4\theta\right) -\cos\left(6\theta\right) $$ I've tried ...
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1answer
109 views

Evaluate $\sum_{k=0}^{511}\frac{\sin\frac\pi{2^{11}}}{\sin\frac{(4k+1)\pi}{2^{12}}\sin\frac{(4k+3)\pi}{2^{12}}}$

I need to evaluate $$\sum_{n=0}^{511}\frac{\sin\frac\pi{2^{11}}}{\sin\frac{(4n+1)\pi}{2^{12}}\sin\frac{(4n+3)\pi}{2^{12}}}$$ Please give me some hint! The final answer is $2^{10}$. By CuriousGuest's ...
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2answers
73 views

Find the angle between the sides 4 and 7 in a right triangle

I need to solve the $B$ corner What I've tried: $$\operatorname{sin} B=\frac47$$ $$B=\operatorname{arcsin}\frac47$$ $$B=34.85$$ But that's not the right answer, can anyone help me find what I did ...
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1answer
203 views

How does the Sin and Cos scale on a slide rule work and what is the formula for it?

As I described in this question, I am trying to make a printable slide rule (similar to the slide rule provided in this SciAm article). I have made most of the parts of it but I can't make the Sin and ...
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1answer
61 views

$\renewcommand{\tan}{\sin}\renewcommand{\arctan}{\arcsin}$Simple trig problem

I have this triangle, I need to find the angle at the $C$ corner. What I tried: $$\operatorname{tan} C=\frac23$$ $$C=\operatorname{arctan}\frac23$$ $$C\approx 33.69$$ However this is the wrong ...
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3answers
229 views

Trigonometric integral evaluation: $\int 4 \sin^4 x \cos^3 x \,dx$ [duplicate]

Evaluate the following integral $$\int 4 \sin^4 x \cos^3 x \,dx$$ I can do simple integration problems, but problems like this seem to stump me, I created this problem so I could solve and compare it ...
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0answers
91 views

Period of trigonometric function

What is the period of $$\frac{7\sin x + 5\cos x}{7\sin{2x} + 11\cos x}$$ What should I do here? I don't even know where to start from. Please help me by giving me a hint!! Thanks.
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1answer
110 views

Evaluate $\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx $

Evaluate the indefinite integral $$\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx$$ I tried multiplying and dividing by $\sec^2 {x} $ and then setting $\tan{x}=y$ but no good. Then I set $\cos ...
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153 views

How can I find $\lim_{x \to 0}\frac{\tan(3x)}{\sin(8x)}$ without L'Hospital's Rule

Is there a way to solve $\lim_{x \to 0}\frac{\tan(3x)}{\sin(8x)}$ without using the trig identity $\tan(3x)=\frac{3\tan(x)-\tan^3(x)}{1-\tan^2(x)}$. I want to know because I had to look up this trig ...
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0answers
50 views

From $\tan(1/A) = \tan(1/B) + \tan(1/C)$ to $A + B + C = ABC$

In this recent question, the equation $$\tan\left(\frac{1}{A}\right) = \tan\left(\frac{1}{B}\right) + \tan\left(\frac{1}{C}\right)$$ is said to imply $$A + B + C = ABC$$ without any stated ...
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3answers
78 views

A Trigonometric Question

If ...
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2answers
94 views

Trigonometric identity on $\cos \pi/7$ [duplicate]

I found this in a book I used to train myself for the Math Olympics a bunch of years ago: Prove that $$\cos\frac{\pi}{7}-\cos\frac{2\pi}{7}+\cos\frac{3\pi}{7}=\frac{1}{2} $$ I couldn't solve it ...
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3answers
65 views

What are the roots of $\sin(ax) + \sin((a + 2)x)$?

I was playing around with $\sin(5x) + \sin(7x)$, wondering where the roots of the function are. I graphed it on wolframalpha and from the list of solutions I guessed that the solutions to $\sin(5x) + ...
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2answers
199 views

Trigonometric identity for $\sin 6x$

I got this question from my teacher: $\sin {6x}=\dots$ Try to make this one from this: $\sin(3x+3x)$, then according to the formula ended up like this: $$2\sin{(2x+x)}\cos{(2x+x)}$$ ...