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

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3
votes
4answers
72 views

Given $\tan A + \tan B = 3x$ and $\tan A \tan B = 2x^2$, find $\tan A - \tan B$ [on hold]

Given $$\tan A + \tan B = 3x$$ and $$\tan A \tan B = 2x^{2}$$ How to find $\tan A - \tan B$ ? I've tried substitution but still couldn't find. EDIT: Can you solve this problem using the formulas for ...
1
vote
3answers
30 views

Graphing of $y= \csc(x)+ \cot(x)$

What's the graph or table of values of $y=\csc(x) + \cot(x)$? I have already solved and graphed the values of $\csc(x)$ and $\cot(x)$.
2
votes
2answers
87 views

Prove that given a triangle satisfying $8\prod \sin\frac{A}{2}=\prod \cos(A-B)$ then that triangle is equilateral.

Prove that given a triangle $ABC$ satisfying $$8 \sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2} = \cos(A-B)\cos(B-C)\cos(C-A)$$ then that triangle is equilateral.
2
votes
3answers
131 views

Trigonometric equation with sine and cosine

So the equation is $3\cos ^2t + 5\sin t = 1$ Now I have simplified this to $$3(1-\sin ^2t) + 5\sin t -1 = 0$$ which leads to $$-3\sin ^2t + 5\sin t + 2 = 0$$ Then I get $$-3t^2 + 5 t +2 = 0$$ Is ...
1
vote
2answers
64 views

In a triangle, find the minimum and maximum of $\cos(A-B)\cos(B-C)\cos(C-A)$

In a triangle, with $A, B, C$ are three angles, find the minimum and maximum of $$\cos(A-B)\cos(B-C)\cos(C-A)$$
4
votes
3answers
198 views

Finding the area of a square that has a circle inside itself

I tried to solve the following problem: I think the image is self-descriptive. I tried to draw a vertical line from the top-end of $\theta$ angle to the horizontal line, then tried to use the ...
3
votes
3answers
68 views

Using the definition of derivative to find $\tan^2x$

The instructions: Use the definition of derivative to find $f'(x)$ if $f(x)=\tan^2(x)$. I've been working on this problem, trying every way I can think of. At first I tried this method: $$\lim_{h\to ...
2
votes
2answers
39 views

Trigonometric equation cos sin and power

The problem is $2\cos t - 3\sin^2t +2 = 0$. I get to $2\cos t -3\sin^2t =-2$ I think that I need to use a trigonometric identity like $\cos(x+y)$ and to divide $2\cos t -3\sin^2t$ with the ...
19
votes
11answers
3k views

What do sine, tan, cos actually mean?

I know that $\sin\theta=\frac{y}{r}$ and $\cos\theta=\frac{x}{r}$. My question is: is $\sin$ a function of $\theta$, as in $\sin (\theta$)? If yes, why is there no $\theta$ on the right hand side of ...
2
votes
2answers
63 views

Solve $\cos3x - 18\cos x +10 =0$

I want to solve $$\cos 3x - 18\cos x +10 =0 $$ I tried: 1) Replacing $\cos 3x$ to $\cos^3x - 3\cos x$ 2) Replacing $\cos x$ to $t$ we get: $$t^3 - 21t +10 = 0$$ So we get cubic equation. But I ...
0
votes
0answers
41 views

Limit approach to infinity [on hold]

During my studying to limits I find this limit but I want to know How we can know that this limit is exist??? $$\lim_{x\to \infty} \sqrt{1-\cos\frac{1}{x} \sqrt{1-\cos\frac{1}{x} ...
2
votes
2answers
34 views

Range of an inverse trigonometric function

Find the range of $f(x)=\arccos\sqrt {x^2+3x+1}+\arccos\sqrt {x^2+3x}$ My attempt is:I first found domain, $x^2+3x\geq0$ $x\leq-3$ or $x\geq0$...........(1) $x^2+3x+1\geq0$ ...
1
vote
5answers
64 views

Simple trigonometry problem

It is given that, $ A+B +C= \pi $,and $\cos A = \cos B \times \cos C$ I have to prove: $\tan B \times \tan C= 2$ to prove that, this is what I did: $$\frac{\sin B}{\cos B} \times \frac{\sin ...
3
votes
3answers
106 views

Trigonometry question

If $$\frac{3-\tan^2\frac{\pi}{7}}{1-\tan^2\frac{\pi}{7}}=\alpha \cos\frac{\pi}{7}.$$ If $\alpha$ is a natural number.Find $\alpha$. My attempt is: ...
0
votes
1answer
43 views

Expressing $\cos(\varphi x)$ as a function of $x\sin\varphi,x\cos\varphi$

Let $\varphi,x\in\mathbb{R}$. I wonder if one can explicitly express $\cos(\varphi x)$ as a function of the variables $x\sin\varphi$ and $x\cos\varphi$. Suppose we denote ...
2
votes
1answer
40 views

Function of sin x

Give that $f(x)=\sin x$ for the domain $0\leq x \leq k$, find the greatest value of $k$ for which $f(x)$ has an inverse. Is the answer $\frac{\pi}{2}$?
1
vote
5answers
105 views

Prove that $\alpha + \beta=\frac {\pi}{2}$

It is given that- (1) $0<\alpha,\beta<90$. (2) $\sin^2\alpha+\sin^ 2\beta=\sin(\alpha+\beta).$ Prove that $\alpha + \beta=\frac {\pi}{2}$
0
votes
1answer
23 views

When does $-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$ where $z$ is a complex variable?

Let $z$ be a complex variable. Is there someone who can show me when does :$$-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$$ Note: I have tried using trigonometric formulas but it didn't work. Maybe I ...
0
votes
1answer
21 views

Deriving the sum to product formula for sine using this method

I am trying to derive $sinC-sinD$ By this method: So far I have tried to set up the same method by beginning with $sin(A+B)-sin(A-B)$, but this reduces to a trivial zero and I can't find another ...
1
vote
1answer
36 views

Determining the position of a polygon inside a circle from only the angle of opposing sides/edges.

For illustration click here I have a simple convex irregular polygon (octagon in example image) inside a circle (circle and polygon are not always concentric and never touching or intersecting) and I ...
-4
votes
0answers
36 views

if sin(x)- cos(x)=4 then find the value of Sec(x)- Tan(x) [closed]

if sin(x)- cos(x)=4 then find the value of Sec(x)- Tan(x)
3
votes
4answers
47 views

Show that $ \tan (A + \theta) $ can be simplified to $- \cot \theta$ as A tends to $\frac{\pi}{2}$

So far I have used the identity, $$\tan\left(\frac{\pi}{2} + \theta\right) = \frac{\tan A + \tan \theta} {1 - \tan A \tan \theta}$$ As $A \to \frac{\pi}{2}$, $\tan A \to \infty$, so my reasoning ...
0
votes
4answers
41 views

What is the equation of a 3D line which represents the intersection between two 3D planes?

The intersection defined by the two planes $v \bullet \begin{pmatrix} 8 \\ 1 \\ -12 \end{pmatrix} = 35$ and $v \bullet \begin{pmatrix} 6 \\ 7 \\ -9 \end{pmatrix} = 70$ is a line. What is the equation ...
0
votes
4answers
54 views

Find the area of the shaded region in the figure

Find the area of the shaded region in the figure What steps should I do? I tried following the steps listed here https://answers.yahoo.com/question/index?qid=20100305030526AAef8nZ But I got 150.7 ...
-2
votes
0answers
16 views

Given the perimeter of an equilateral triangle, and a fraction of the perimeter, how would i find the (x,y) coordinates of the triangle.

Given the perimeter of an equilateral triangle, and a fraction of the perimeter, how would I: Create a formula for finding the x coordinate Create a formula for finding the y coordinate of the ...
11
votes
6answers
560 views

Wanted : for more formulas to find the area of a triangle?

I know some formulas to find a triangle's area, like the ones below. Is there any reference containing most triangle area formulas? If you know more, please add them as an answer ...
3
votes
1answer
37 views

I dont see how this algebraic manipulation is valid (trig functions)

So they have two equations: $v_{x}=V_0 \cos\theta-2\Omega V_o \sin\lambda \sin \theta *t$ $v_{y}=-V_0 \sin\theta -2 \Omega V_o \sin\lambda \cos \theta *t$ And they say "to lowest order in $\Omega$, ...
0
votes
8answers
56 views

Prove the following [Trigonometric Equation] [closed]

The equation to be proved is : $$\frac{\cot A \cos A }{\cot A + \cos A} = \frac{\cot A - \cos A} {\cot A \cos A}$$ I have tried taking both LHS and RHS to solve it, but I have not been able to solve ...
3
votes
10answers
105 views

Point of intersection of $f(x)=\sin(2x)+\cos(2x)$ and the $x$-axis

How can I algebraically (without looking at the graph) find the point of intersection of $f(x)=\sin(2x)+\cos(2x)$ and $x$-axis, in the interval $[0, \pi]$?
1
vote
1answer
50 views

Can you help me with a function with cos and radians?

I have this function: $$d(n)=15-2,5\cdot\cos\left(\pi\frac{n-31}{360}\right).$$ This function talks about the difference between when day start and day over. When I change $n$ for $5$ result for me ...
6
votes
3answers
62 views

Find the sum: $\sum_{i=1}^{n}\dfrac{1}{4^i\cdot\cos^2\dfrac{a}{2^i}}$

Find the sum of the following : $S=\dfrac{1}{4\cos^2\dfrac{a}{2}}+\dfrac{1}{4^2\cos^2\dfrac{a}{2^2}}+...+\dfrac{1}{4^n\cos^2\dfrac{a}{2^n}}$
-1
votes
3answers
34 views

trigonometry circles and arcs [closed]

$AB$ is a diameter of a circle whose center is $O$. $P$ is a point on the circumference such that the chord $AP=8\ cm$ and $BP=6\ cm$. Calculate the values of $\angle PAB$ and $\angle POB$ in ...
0
votes
0answers
26 views

$S= \frac{1}{\cos x \cos2x} + \frac{1}{\cos2x\cos3x} + \frac{1}{\cos3x \cos4x} + \cdots + \frac{1}{\cos(n-1)x \cdot \cos{nx}}$ [duplicate]

Compact $S= \dfrac{1}{\cos x\cos2x} + \dfrac{1}{\cos2x\cos3x} + \dfrac{1}{\cos3x\cos4x} +...+ \dfrac{1}{\cos(n-1)x\cos{nx}}$
0
votes
4answers
35 views

Triangles: find unknown distance

To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle $75°$ from the horizontal. An observer $D = 500 m$ away measures the angle of elevation to the ...
2
votes
1answer
30 views

Fundamental definition of trigonometric functions

What is the most basic or fundamental definition of a trigonometric function, (say sine)? How is sine of an angle defined? I looked up on wikipedia, and it seems that sine of an angle stems from this ...
2
votes
1answer
35 views

What is the number of distinct elements in $S$?

Allow for these values: $$A = \begin{pmatrix} \cos \frac{2 \pi}{5} & -\sin \frac{2 \pi}{5} \\ \sin \frac{2 \pi}{5} & \cos \frac{2 \pi}{5} \end{pmatrix} \text{ and } B = \begin{pmatrix} 1 ...
0
votes
0answers
24 views

Characteristic polynomials for matrix A, involving the Identity matrix

Let us say we have a square matrix A, where A's characteristic polynomial is defined as $P_A(t) = \det (t I - A)$ (In this problem, I represents the identity matrix which has the same dimensions as ...
0
votes
3answers
99 views

Why is $\tan^{-1}(0)=0$?

My calculator tells me that $\tan^{-1}(0)=0$. If $\tan^{-1}(\theta)=\frac{\cos\theta}{\sin\theta}$ and $\cos0=1$ and $\sin0=0$, then $\tan^{-1}(0)=\frac{\cos0}{\sin0}=\frac{1}{0}$ and should be ...
1
vote
1answer
92 views

Find A such that $A^2 \neq I$ but $A^4 = I$ [duplicate]

Find a $3 \times 3$ matrix A such that $A^2 \neq I$ but $A^4 = I$, where $I$ is the $3 \times 3$ identity matrix. Is there a simpler way to solve this problem rather than bashing it out by ...
2
votes
2answers
96 views

Finding $4$ variables using $3$.

if I have: $ x=\dfrac{a-.5b-.5c+.25d}{a+b+c+d}$ $ y=\dfrac{\dfrac{b\sqrt{3}}{2}+\dfrac{c\sqrt{3}}{2}+\dfrac{d\sqrt{3}}{4}}{a+b+c+d}$ $ z=a+b+c+2d $ Then how do I get back to: $ a= $ , $ b= $ , $ ...
0
votes
3answers
26 views

Triangle, angle of depression and elevation: find unknown distance

A woman standing on a hill sees a flagpole that she knows is $35$ ft tall. The angle of depression to the bottom of the pole is $14^{\circ}$, and the angle of elevation to the top of the pole is ...
0
votes
1answer
21 views

Solving trigonometric functions [duplicate]

I am having trouble finding the solution to this equation. I keep getting a negative answer when I know it must be positive as t is time in the problem. I want to solve $d(t)= 3.5\cos((\pi/6)t)+4.5$ ...
2
votes
1answer
39 views

Determining exact solutions of $10 \sec \theta + 2 = -18$

I am trying to determine the exact solutions for the angle theta, where $0 \le \theta < 2\pi$ The equation I have been given is $$10\sec\theta+2=-18$$ I am having trouble with this question as I ...
4
votes
3answers
86 views

The exact value of $\cot\frac{7\pi}{6}$?

I am working on a trigonometry question at the moment and am very stuck. I have looked through all the tips to solving it and I cant seem to come up with the right answer. The problem is What is ...
2
votes
3answers
77 views

Prove that $ \tan40° + \sqrt 3 =4 \sin40° $

The equality I'm trying to prove looks like that: $$ \tan40° + \sqrt 3 =4 \sin40° $$ My guess is that $\sqrt3$ can be rewritten as $\tan60°$ and I can use proved in previous exercise formula $$\tan3 ...
2
votes
2answers
42 views

Calculating values of $1 - \cos(x)$ for $x$ near zero using computer arithmetic

Explain why calculating values of $1 - \cos(x)$ where $x$ near zero using the trigonometric identity $1 - \cos(x) = 2\sin^2\big(\frac{x}{2}\big)$ will result in more accurate results. Is it because ...
0
votes
4answers
93 views

How to simplify $\sin^4 (x)$?

Folks, how can $\sin^4(x)$ be simplified to a trig function with power of 1? I tried: $(\sin^2(x))^2 = ((1-\cos2x)/2)^2$ but still getting $\cos x$ to the power of $2$. Wolframalpha only shows the ...
2
votes
5answers
51 views

Is the substitution of standard angles while proving the equality of trigonometric formulas allowed?

Here is a problem that my class 10 maths teacher gave me: Prove that $\sec^4\theta$ - $\sec^2\theta$ = $\tan^4\theta$ + $\tan^2\theta$ She expected me to use trigonometric identities to prove ...
2
votes
3answers
73 views

Is there no such identity as $\csc^2+\sec^2=1$?

$$\csc^2+\sec^2=1?$$ I thought I could just use reciprocal from the other formula $\sin^2+\cos^2=1$, can you explain what's wrong?
0
votes
0answers
34 views

solution for triangles [closed]

A surveyor wishes to find the width of a stream without crossing it. He measures a line $CB$ along the bank, $C$ being directly opposite to a point $A$ on the farther bank ($\angle ACB= 90^\circ$)the ...