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

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8
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2answers
241 views

Prove $\tan 54^\circ=\frac{\sin24^\circ}{1-\sqrt{3}\sin24^\circ}$

How to prove this identity without using the actual values of $\tan54^\circ$ and $\sin24^\circ$ $$\tan 54^\circ=\dfrac{\sin24^\circ}{1-\sqrt{3}\sin24^\circ}$$ Edit: I still don't get it, I am ...
-1
votes
2answers
4k views

Find the exact values without using a calculator of cos^-1(-1/2), tan^-1(-√3/3) and sec^-1 (2)

How do I solve this problems? The inverse of cosine is secant and the the inverse of tan is cotangent and the inverse of secant is cosine. Is that how I should think of it?
0
votes
2answers
883 views

Find the exact values without a calculator: (a) $\tan \frac{11\pi }6$ (b) $\sec \frac{-3\pi}4$ (c) $\cot \frac{-5 \pi}3$

Okay I know the unit circle back and forth, but I get confused when I am asked to find answers that do not refer to sine and cosine. For example, I am ask to evaluate $\tan \frac{11\pi }6$. Since ...
0
votes
0answers
80 views

How prove $\tan1^{\circ}+\tan7^{\circ}+…+\tan175^{\circ}=-30\sqrt3$?

How prove that $\tan1^{\circ}+\tan7^{\circ}+...+\tan175^{\circ}=-30\sqrt3$? I noticed that $1^{\circ},7^{\circ},...,175^{\circ}$ form an arithmetic.
-2
votes
3answers
65 views

Trigonometric Solution.

I am confused with the different methods of finding tirg solutions and always ending up getting a different and often wrong answer from the real one. For example: $cot\theta +cosec\theta =\sqrt{3}$ ...
0
votes
1answer
60 views

Triangle question, proving isosceles given trigonometric conditions

$ABC$ is a triangle satisfying the following condition: $$\frac{\sin B}{\sin A}=\frac{\tan B+\cot C}{\tan A+\cot C}$$ How do I prove that $ABC$ is isosceles? I really have no idea.
1
vote
2answers
33 views

Trigonometric Identity Subtraction

I can't do problems such as $ \displaystyle \csc \theta - \sin \theta = \frac{\cos^2 \theta}{\sin \theta}$ Because I simply do not know how to deal with the 'subtraction' component involved. I don't ...
1
vote
3answers
110 views

Trigonometric identity expressing $\sec \theta+\text{cosec } \theta$ in terms of sine and cosine

$\large{\text{cosec }\theta+\sec{\theta}=\dfrac{\sin\theta+\cos\theta}{\sin\theta\,\cos\theta}}$ I know that cosecant is the inverse of sine, and secant is the inverse of cosine. However, that ...
3
votes
5answers
258 views

Proof of trigonometric identity $\cot \theta \sec\theta= 1/ \sin\theta$

Is this trigonometric identity provable? $$\color{red}{}\;\color{navy}{\cot \theta \sec \theta = \dfrac 1 {\sin \theta}}$$ I can't seem to get passed: $\dfrac{1}{\tan\theta \cos\theta}$
1
vote
3answers
48 views

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? ...
3
votes
2answers
83 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 ...
4
votes
6answers
178 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 ...
0
votes
2answers
100 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} = ...
1
vote
1answer
40 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 ...
0
votes
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 ...
0
votes
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 ...
8
votes
4answers
905 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 ...
0
votes
3answers
51 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 ...
0
votes
1answer
293 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.
4
votes
6answers
505 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...
1
vote
4answers
342 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 ...
0
votes
2answers
57 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 ...
3
votes
1answer
110 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: ...
0
votes
4answers
186 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 ...
1
vote
3answers
39 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: $(m\cos\alpha-n\sin\alpha)^{2}=p^2$ ...
1
vote
2answers
141 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 ...
3
votes
1answer
274 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 = ...
3
votes
1answer
381 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 ...
0
votes
4answers
68 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 ...
1
vote
2answers
1k 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 $\sin \theta = \frac{12}{13}$ ...
0
votes
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 ...
1
vote
3answers
903 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 ...
2
votes
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 ...
0
votes
1answer
62 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 $$ \cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)? $$
1
vote
2answers
46 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 ...
0
votes
3answers
72 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 ...
1
vote
1answer
377 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 ...
2
votes
1answer
201 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 ...
-1
votes
2answers
201 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.
1
vote
2answers
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 ...
5
votes
3answers
67 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: ...
-1
votes
1answer
51 views

Question about writing cyclometric function in function of x

I have an excercise about cyclometric functions and I'm stuck right now: $\cot(2*\operatorname{arcsec}(x))$ Let $ y=\operatorname{arcsec}(x) \Leftrightarrow \sec(y)=x$ then $\cot(2y)=\frac ...
2
votes
1answer
120 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! :)
2
votes
2answers
50 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 $$ ...
1
vote
2answers
159 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 ...
2
votes
1answer
85 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 ...
1
vote
1answer
110 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 ...
1
vote
2answers
75 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 ...
2
votes
1answer
229 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 ...
1
vote
1answer
63 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 ...