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
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2answers
64 views

Elementary Trigonometry problem

My problem is from Israel Gelfand's Trigonometry textbook. Page 9. Exercise 7: Two points, $A$ and $B$, are given in the plane. Describe the set of points $X$ such that $AX^2+BX^2=AB^2.$ The ...
0
votes
2answers
87 views

What is the meaning of calculating sine of a number?

When we calculate sine/cos/tan etc. of a number what exactly are we doing in terms of elementary mathematical concept, please try to explain in an intuitive and theoretical manner and as much as ...
1
vote
2answers
790 views

solve $\tan(x) = \sqrt{1-x^2}$

I am not sure if you should be deriving it or converting tan into $\sin(x)/\cos(x)$. Even then, I do not know what to do from there
1
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1answer
76 views

Find the 6th root of $-3+4i$ and plot on complex plane

So I have a rough idea on how to get the answer but I'm getting stuck on the angle or argument for the equation. The question is: Find the 6th root of $-3+4i$. I first find the $r$ value which ...
1
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2answers
473 views

Solving for Cos Exactly

How to solve $\cos(\dfrac{5\pi}{6})$ and $\cos(\dfrac{7\pi}{6})$ exactly? I couldn't use special triangles to solve this either.
2
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2answers
73 views

Get the angle in a circle using center, radius and one point in a circle.

There is a circle and i know Point1 this is fixed and i know another point Point2 which can be anywhere in the circle. and i want to know the angle which is made at center. Thanks Your help will be ...
1
vote
2answers
86 views

How to solve equations containing trigonometric functions and powers

I mean, is there a way to solve analytically something like this: $$ \tan(x) - x = 0 $$ or like this equation $$ \tan(x) - x^2 = 0 $$ I know this will produce infinite number of roots but could ...
0
votes
1answer
165 views

Value of $\frac{\cos 45}{\sec 30 + \operatorname{cosec} 30}$

I just put the values from the trignometric table to solve, but the answer is different in the answer book. $$\frac{\cos 45}{\sec 30 + \operatorname{cosec} 30}$$
0
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4answers
69 views

Prove that $1+\tan^2 x=\sec^2 x$ [duplicate]

I have no idea how to prove this. Does anyone know where to start? We're allowed to use other trigonometric identities but i'm not sure why these are useful.
1
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2answers
202 views

Basic trigonometry identities question

$$\eqalign{\tan^2\theta-\sec^2\theta &=\tan^2\theta-\dfrac1{\cos^2\theta}\\&=\dfrac{\sin^2\theta}{\cos^2\theta}-\dfrac1{\cos^2\theta}\\&=-\dfrac{\cos^2\theta}{\cos^2\theta}\\&=-1.}$$ ...
1
vote
2answers
2k views

Is there an ambiguous case for cosine law. Why or why not.

A really specific answer would be helpful.
1
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3answers
48 views

Solve for Radian Exactly

$$\tan(A) = \frac{\sqrt{3}}{-3}$$ I've tried using special triangles but couldn't find a matching faction using sohcahtoa.
0
votes
2answers
32 views

Half tangent representation

If $x$ is defined by the interval $\pi/2>x>0$, and $\tan(x)=A$, what is $\tan(x/2)$? This is a multiple choice question on a test, and I don't have a approach because all the answer choices are ...
1
vote
1answer
22 views

Trig expression simplification

Could someone explain how to simplify $\dfrac{\sin(2x)}{2-2\cos^2(x)}$? I've had tried the power reduction identity but the result did not seem much more simple. Any help would be appreciated.
2
votes
1answer
84 views

Do sine and cosine of complex numbers have anything to do with right-triangles or circles?

I've recently been working on a web application that draws iterating function generated fractals. I've noticed that the sine and cosine functions can be used to draw exquisite plots using an ...
1
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1answer
89 views

finding exact value of $\sec^{-1} 5$

Find the exact value of $\sec^{-1} 5$ (decimal answer). I know that $\sec^{-1}5=\cos^{-1}\dfrac{1}{5}$, but I don't know how to proceed from here. I drew a right triangle with sides $1$ and $5$ ...
0
votes
2answers
72 views

Period of $\frac{\sin(Ny)}{\sin y}$ with $N$ odd?

The function $$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$ is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ ...
1
vote
2answers
65 views

Could someone explain me this “ownership” of the arctangent [duplicate]

someone could explain to me this: $$\int { \arctan { \left( \frac { 1 }{ { u }^{ 2 } } \right) } } \,du=\int { \frac { \pi }{ 2 } } -\arctan { \left( { u }^{ 2 } \right) } \, du$$
1
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2answers
60 views

Substitution of an implicit variable

I wasn't sure how to title this question: I want to manipulate the integral $$I(a,b) = \int_0^{\frac{\pi}{2}} \frac{d \phi}{\sqrt{a^2\cos^2 \phi + b^2 \sin^2 \phi}}$$ with this subsitution: $$\sin ...
0
votes
1answer
178 views

Trigonometry / Obtuse angle

If $\cos A = 4/5$ and $\sin B = 5/13$, where $A$ is a acute and $B$ is obtuse, find, without evaluating the angles $A$ and $B$, the values of a) $\sin (A-B)$ b) $\cos (A+B)$ I'm stuck figuring out ...
1
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2answers
57 views

Trigonometry / Finding the exact value

Given that $\cos \theta = \dfrac{-4}{5}$ and $\sin \theta$ is positive, obtain the exact values of $\cos (6\pi+\theta)$ i don't understand this question.
1
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2answers
73 views

Derivative of a Trigonometric Function Help

Trying to derive a trigonometric function, Wolfram Alpha and my textbook provide two different answers. Here is the function: $$y = {\cot x\over (1+\csc x)}$$ First step using quotient rule results ...
1
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2answers
94 views

Help finding the second derivative of this function.

I need help finding the second derivative of this function. I found the first derivative and the second, but the program says my answer is incorrect either by typing error and it won't process ...
0
votes
2answers
79 views

Trigonometry reference angle of radian

Given that $-2\pi≤\theta≤0$ and $\theta$ has a reference angle of $\cfrac{\pi}{6}$ , find $\theta$ if it is in the a) 1st quadrant b) 2nd quadrant c) 3rd quadrant d) 4th quadrant I need help on ...
0
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2answers
142 views

Converting angles in Radians to degrees

convert following angles to degrees. Give your answer correct to 2 decimal points. a) $-3.5$ $-3.5 \times \dfrac{180}{π} = ?$ i'm stuck on this stage..
0
votes
1answer
51 views

Is this calculation correct? and why?

1st - is my value correct 2nd - Why or why not? I have a regular hexagon, which i know the height is 2 units and width is also 2 units. I am trying to find the value for one of the sides. I broke ...
0
votes
3answers
80 views

Why is $\sin 150^o\times\sin 20^o=\sin 80^o\times\sin 10^o$

I encountered this calculation in a problem $\dfrac{\sin 150^o\times\sin 20^o}{\sin 80^o\times\sin 10^o}$ and calculated that it equals 1. Is it just a coincidence or is there any identity that says ...
0
votes
1answer
64 views

Find points near end point of a line

Any equation to find points near to both start and end points of lines with different slopes. See image. Need P and Q. If Endpoints are named A and B, AP and BQ should be 1 cm
0
votes
3answers
86 views

How does $cos(x) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}| \cdot |\vec{w}|}$ make sense?

In Multivariable Calculus, the professor said that in order to compute the angle $x$ between two vectors $v$ and $w$, we use the formula: $cos(x) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}| \cdot ...
1
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1answer
34 views

Solutions for trigonometric equation

Consider the following equation: $$ 2 = \cos(x) + \cos(\sqrt{2}x) $$ So, I'm reading this proof which says that the equation has a solution iff $$ x=0 \mod{2\pi} \quad \wedge \quad \sqrt{x}=0 ...
0
votes
1answer
19 views

Trigonometry involving usage of series

What is the value of $[\cos(\pi)/2n + 1] + [\cos(3\pi)/2n + 1] + \cos(5\pi)/2n + 1]$ ..... upto $'n'$ terms equals? Can anyone, please solve this question for me?
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2answers
52 views

Express the length of the as a function of x

I am having problems understanding how to extract this information into a formula. ...
1
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0answers
35 views

$\frac {1 } {10 }(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2)^2+(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1)^2) \le (y_1-x_1)^2+(y_2-x_2)^2$?

Is it true that: $$\frac {1 } {10 }\left(\left(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2\right)^2+\left(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1\right)^2\right) \le (y_1-x_1)^2+(y_2-x_2)^2$$ I think I should ...
0
votes
2answers
48 views

If $y=2\sin^{-1}\sqrt{1-x}+\sin^{-1}(2\sqrt{x(1-x)})$ for $0<x<\displaystyle\frac{1}{2}$ then what is the value of $\displaystyle\frac{dy}{dx}$

If $y=2\sin^{-1}\sqrt{1-x}+\sin^{-1}(2\sqrt{x(1-x)})$ for $0<x<\displaystyle\frac{1}{2}$ then $\displaystyle\frac{dy}{dx}$ equals : A) $\displaystyle\frac{2}{\sqrt{x(1-x)}}$ B) ...
1
vote
4answers
118 views

Finding $\sin^{-1}(x)$ without using a calculator

I don't understand how to compute $\sin^{-1} (0.6293)$, to figure out the angle without using a calculator. I understand how to find the answer if I use a calculator but I don't understand the ...
1
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0answers
204 views

Horizontal axis for reference angles

Why we always take the horizontal axis for reference angles? Is it by convention? Could it have been the y-axis? What advantages do we gain from taking the horizontal axis?
0
votes
1answer
54 views

I need help on deriving this trig function.

I am trying to derive $e^x \sin x - 2x \csc x$. I tried using the product and difference rule. So I got the derivative for $e^x \sin x$ and got $(e^x)(\cos(x))+(\sin(x))(e^x)$ and for the derivative ...
0
votes
3answers
48 views

Prove this trigonometric identity…

$$\sin({495}^{\circ})-\sin({795}^{\circ})+sin({1095}^{\circ})=0$$ So I have to prove that the identity is correct. How can I transform those large angles in smaller ones?
1
vote
2answers
197 views

Trigonometric ratios involving negative angles

Given that $\cos \theta = \dfrac{3}{5}$ and csc is positive a) which quadrant is $\theta$ in? Hence deduce the quadrant that $-\theta$ is in. so for question it is 2nd quadrant b) Without finding ...
0
votes
1answer
57 views

Trigonometry / reference angle

My question is how to find the reference angle for the following angle? Θ=7π/6 what i get was $\dfrac{7\pi}{6}-\dfrac{6\pi}{6}= -\dfrac{11\pi}{6}$ i'm still unsure im doing it correctly
0
votes
3answers
86 views

Trigonometry: Find $\sin \theta$ when $\tan \theta$ is known.

if $\tan \theta = \sqrt{63}$ and $\cos \theta$ is negative, find $\sin \theta$. So since $\tan \theta$ is positive and $\cos \theta$ is negative, it lies in the $3$rd quadrant. So $\sin$ is ...
0
votes
1answer
68 views

Finding bounding rectangle after rotation

I have two rectangles inner rectangle(green) and the outer one (red). OuterRectangle will be calculated by adding some offset to inner rectangle as shown below. OuterLeft = innerLeft - 100; ...
2
votes
2answers
51 views

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.
1
vote
1answer
76 views

Providing solutions for the intersection of two trigonometric functions

I'm trying to find a general solution for the intersection of two trigonometric functions: $$a(x)=500\sin \left( \frac{\pi x}{2} \right)+150$$ $$a(x)=200\cos \left( \frac{\pi }{2}\left( x+\frac{\pi ...
1
vote
4answers
819 views

how can I find the period of $f=\sin^4(x)$?

How can I find the period of $f=\sin^4(x)$? $\sin^4(x)=\frac{3-4\cos2x+\cos4x}{8}$. I didn't manage to reduce $\cos 4x$ to $\cos 2x$.
0
votes
1answer
75 views

Proving Sin Cos Tan

I am asked to prove the following: $$\dfrac{1-\cos x}{\sin x}=\dfrac{\sin x}{1+\cos x}=\tan\dfrac x2.$$ Looking at the answer I am not able to see what is going on here: $$\frac{1 - ...
1
vote
0answers
32 views

Is this solution correct

Given $A+B=\frac{\pi}{4}$, find $(1+\tan A)(1+\tan B)$ My attempt: Since $\tan(A+B)=1=\frac{\tan(A)+\tan(B)}{1-\tan(A)\tan (B)}$, therefore $\tan(A)+\tan(B)+\tan(A)\tan(B)=1$, therefore, ...
5
votes
7answers
842 views

Finding all the values of $\theta$ for which $\tan(\theta)=\sqrt3$; problem with understanding.

My textbook has a section where it says a possible way that $\tan(\theta)$ can be thought of is: For acute angles $\theta$, $\tan(\theta)$ is the $y$-coordinate of the point on the terminal side ...
2
votes
4answers
113 views

integration by parts of trig functions

Can anyone help me with this integral? $\int{x^3 \sin(x^4) dx}$ I set $u=x^3$, and I let $v=-\cos(x^4)$, so that $\frac{dv}{dx}=\sin(x^4)$ I tried using integration by parts, but, whenever I come ...
1
vote
2answers
79 views

Help in Solving a Trigonometric Equation

Solve the equation $$\left(\sin x + \cos x\right)^{1+\sin(2x)} = 2$$ when $-\pi \le x \le \pi $ . I have tried to use $\sin (2x) = 2\sin x \cos x$ identity but I this doesn't lead me to a conclusion. ...