Tagged Questions
0
votes
0answers
15 views
Find the terminal point when the distance is not in terms of $\pi$
From Stewart Precalculus 5th edi, P407
I am not sure what to do here, in the textbook, Steward didn't provide any example as to finding the terminal point when the distance $t$ is an integer. I ...
3
votes
3answers
72 views
Determining $\sin(15)$, $\sin(32)$, $\cos(49)$, etc.
How do you in general find the trigonometric function values? I know how to find them for 30 45, and 60 using the 60-60-60 and 45-45-90 triangle but don't know for, say $\sin(15)$ or $\tan(75)$ or ...
4
votes
2answers
71 views
Trigonometry Airplane question. Finding bearing and distance.
A little background(if you don't care for my story, skip straight to the question): I've missed a few lectures from my teacher because I fell ill. Since I have no information to work with other than ...
5
votes
2answers
162 views
If ${ x }^{ 4 }+{ y }^{ 2 }=1$ then $x$ and $y$ can be both rational numbers?
Can you give two numbers $(x,y)\in\mathbb{Q}$ such that ${ x }^{ 4 }+{ y }^{ 2 }=1$.
I don't know if exists or not. I derive this equation questioning that if $\sin { \alpha } ={ x }^{ 2 }$ for ...
1
vote
1answer
74 views
$\pi$ is just a number, or also the circumference of a sub-unit circle?
A unit circle defined in the Cartesian plane has a radius of $1$ and a diameter of $2$. So making a full round is $2 \pi$. Now, $\pi$ is the ratio of the circumference over the diameter, so if I have ...
1
vote
2answers
36 views
Manually Finding Values of Inverse Trigonometric Functions
I'm trying to solve (for $x$) some problems such as $\arctan(0)=x$, $\arcsin(-\frac{\sqrt{3}}{{2}})=x$, etc.
What is the best way to go about this? So far, I have been trying to solve the problems ...
2
votes
2answers
67 views
How can I prove this cosine equation?
How to prove that $\cos(90)\cos(\theta)+\sin(90)\sin(\theta)=\sin(\theta)$ ?
1
vote
2answers
60 views
How to simplify $\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $
How to simplify the following expression :
$$\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $$
1
vote
2answers
54 views
Determining pendulum rise using trigonometry
Everyone in my math class (including the teacher) is having problems with this trigonometry question:
I am assuming that you halve the pendulum and the bottom of the triangle would be $\frac{1.8}{2} ...
2
votes
3answers
107 views
Limit without L'Hopital's rule
Please solve this withouth L'Hopital's rule? $$\lim_{x\rightarrow\sqrt{3}} \frac{\tan^{-1} x - \frac{\pi}{3}}{x-\sqrt{3}}$$
All I figured out how to do is to rewrite this as $$\frac{\tan^{-1} x - ...
0
votes
1answer
55 views
Is my technique valid? [duplicate]
I have serious doubts about this, but I thought you guys might at least fix this and suggest something useful which would make this approach work. The question is to prove that $$2=2\cos(x)+x\sin(x)$$ ...
-2
votes
2answers
35 views
Calculus help needed.
In this question you must demonstrate knowledge of derivatives and the use of the differentiation rules, as well as basic algebra. Hence, you must show your working.
Any help is much appreciated!
3
votes
1answer
34 views
Trig and algebra problem: Finding sides of a triangle
Let $ABC$ be a triangle such that $\angle ACB = \pi/6$ and let $a,b,c$ denote the lengths of the sides opposite to $A,B,C$, respectively. What are the value(s) of x for which $a = x^2 + x + 1, b = ...
1
vote
2answers
43 views
What is the best way to solve equations with trig functions
I usually use guessing to solve equations with trig functions. Yesterday, I came across an equation that I couldn't really write it in a helpful form to guess. My question is, how can I solve equation ...
0
votes
1answer
39 views
Finding a function which fits this data?
I need to find a polynomial (or other continuous elementary function) on the interval [70, 180] such that it passes through the points (70, 0) (this is a relative min), (105, 17) (this is a relative ...
0
votes
3answers
27 views
steps for finding inverse tangent
My textbook gives me this:
Can someone please walk me through the steps to get -pi/3 from the inverse tangent?
1
vote
1answer
42 views
Adding integer multiples of pi
I have an angle with a given radian measurement and need to express it differently by adding integer multiples of pi. Is it accurate to say that I can simply add 4 to the coefficient of pi? It seems ...
5
votes
1answer
88 views
Expressing a number in $\sqrt a/b$ form
Express the number $\sqrt3 \sin(10^\circ) +\dfrac38\tan(10^\circ)$ in the form $\dfrac{\sqrt a} b$, where $a$ and $b$ are integers.
I am sure that trigonometric formulas must be used here, but I ...
1
vote
1answer
19 views
Formula for Damage Calculation Based on Distance
I am working on programming a game, and I want the damage that a player takes to be based off of the distance they are from the explosive. Consider it like a grenade. Right on the blast radius ...
1
vote
1answer
41 views
If $A+B+C=\pi$ does it imply that $\sin2A+\sin2B+\sin2C=4\sin A\sin B\sin C$ [duplicate]
If $A+B+C=\pi$ does it imply that $$\sin2A+\sin2B+\sin2C=4\sin A\sin B\sin C$$
If yes, how?
0
votes
1answer
24 views
Solution of two equivalent quadratic equation solutions
I have two solutions to quadratic equations, based on the quadratic formula. The solutions are equivalent. Additionally, one of the variables (Tx, ...
0
votes
1answer
75 views
De Moivre's theorem question.
State De Moivre's theorem and use it to find integers $ A,B,C$ such that $$\sin^5\theta=A\sin5 \theta + B\sin3\theta + C\sin\theta.$$
I know De Moivre's theorem, how to prove it, and converting to ...
7
votes
3answers
135 views
Find this $a,b,c$ such that $\sqrt{9-8\sin 50^{\circ}}=a+b\sin c^{\circ}$
It is known that$$\sqrt{9-8\sin 50^{\circ}}=a+b\sin c^{\circ}$$
for exactly one set of positive integers $(a,b,c)$ where $0<c<90$
find the value
$$\dfrac{b+c}{a}$$
my idea,$ \sin 50^\circ ...
3
votes
1answer
107 views
Largest element of the set $\{ \sin{1}, \sin{2}, \sin{3}\}$
i have to find the largest element of the following set $\{ \sin{1}, \sin{2}, \sin{3}\}$.
I converted every element to the first quadrant so i can use the monotony of cosine, the set becomes:
...
4
votes
1answer
90 views
Find area bounded by two unequal chords and an arc in a disc
Math people:
This question is a generalization of the one I posed at Find area bounded by two chords and an arc in a disc . Below is an image of a unit circle with center $O$. $\theta_1, \theta_2 ...
1
vote
1answer
44 views
A finite sum of trigonometric functions
By taking real and imaginary parts in a suitable exponential equation, prove that
$$\begin{align*}
\frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases}
1&\text{if } k ...
3
votes
3answers
107 views
Show that $\tan {\pi \over 8} = \sqrt 2 - 1$
Show that $\tan {\pi \over 8} = \sqrt 2 - 1$, using the identity $\tan 2\theta = {{2\tan \theta } \over {1 - {{\tan }^2}\theta }}$
Using $\tan 2\theta = {{2\tan \theta } \over {1 - {{\tan ...
0
votes
1answer
50 views
Angle between 2 faces of a pyramid
The problem:
Given a pyramid with $P_0=(0,0,0)$, $P_1=(1,1,1)$, $P_2=(2,-1,2)$, $P_3=(3,0,1)$, find the angle between the $P_1P_2P_3$ face the $P_0P_1P_2$ face.
My idea for the solution is to ...
1
vote
3answers
52 views
How do I prove that $2\cos (2\theta + {\pi \over 3}) \equiv - 2\sin(2\theta - {\pi \over 6})$
Using the identity $\cos (\theta + {\pi \over 2}) \equiv - \sin\theta $
4
votes
7answers
231 views
How do I prove: $\cos (\theta + 90^\circ) \equiv - \sin \theta $
How do I go about proving this?
I know one method is:
$\eqalign{
\cos (90^\circ + \theta ) &\equiv \cos90^\circ \cos\theta - \sin90^\circ \sin\theta \cr
& \equiv (0)(\cos\theta ) - ...
1
vote
4answers
156 views
Find the value of $\alpha $ given $2\sin\theta -\sqrt 5 \cos \theta \equiv - 3\cos (\theta + \alpha )$
Given:
$$2\sin\theta -\sqrt 5 \cos \theta \equiv - 3\cos (\theta + \alpha ),$$
where $$0 <\alpha < 90^\circ, $$
find $α.$
The issue I have with this question is the $-3$ on the right hand ...
3
votes
1answer
50 views
Trigonometry - Calculating the pyramid volume
The problem:
There be the points $P_0(0,0,0)$, $P_1(1,1,1)$, $P_2(2,-1,2)$ and $P_3(3,0,1)$. Calculate the volume of the pyramid.
Now I assumed the base of the pyramid is a triangle, with points ...
4
votes
0answers
195 views
Find area bounded by two chords and an arc in a disc
Math people:
Let $D$ denote the unit disc in $\mathbb{R}^2$ with center $(0,0)$. Let $\theta \in (0, \pi)$, and let $U \subset D$ be the horizontally symmetric region obtained by removing from the ...
5
votes
2answers
179 views
$ \sin^{2000}{x}+\cos^{2000}{x} =1$ equation explanation
Solve the equation:
$$ \sin^{2000}{x}+\cos^{2000}{x} =1.$$
What I did:
$\sin^2{x} =1 \land \cos^2{x}=0$ when $x=\frac \pi2 + \pi k $
$\cos^2 {x} =1 \land \sin^2{x}=0$ when $x= \pi k$
I think ...
0
votes
2answers
47 views
Intersection between a line and a wave
Is it possible to find the points of intersection between a line and a sine wave? I would like a function to find the nth intersection, rather than just first intersection within the domain of the ...
3
votes
1answer
117 views
What does it mean to eliminate $\theta$ from these equations? How should I do it?
Eliminate $\theta$ from the following pairs of equations:
A) $x=\sin \theta$, $y=\sin 2\theta$
B) $x=3\cos 2\theta +1$, $y=2\sin \theta$
My problem is I really don't understand what ...
3
votes
2answers
139 views
Proving $\tan (\frac{\pi }{4} - x) = \frac{{1 - \sin 2x}}{{\cos 2x}}$
How do I prove the identity:
$$\tan \left(\frac{\pi }{4} - x\right) = \frac{{1 - \sin 2x}}{{\cos 2x}}$$
Any common strategies on solving other identities would also be appreciated.
I chose to ...
6
votes
8answers
258 views
Solving $\sin \theta + \cos \theta=1$ in the interval $0^\circ\leq \theta\leq 360^\circ$
Solve in the interval $0^\circ\leq \theta\leq 360^\circ$ the equation $\sin \theta + \cos \theta=1$.
I've got the two angles in the interval to be $0^\circ$ and $90^\circ$, it's not an answer I'm ...
3
votes
5answers
192 views
Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$
$$\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$$
I have attempted this question by expanding the left side using the cosine sum and difference formulas and then multiplying, and then ...
9
votes
3answers
191 views
Being ready to study calculus
Some background: I have a degree in computer science, but the math was limited and this was 10 years ago. High school was way before that. A year ago I relearnt algebra (factoring, solving linear ...
-2
votes
5answers
145 views
$\cos(\arcsin(x)) = \sqrt{1 - x^2}$. How?
How does that bit work?
How is
$$\cos(\arcsin(x)) = \sin(\arccos(x)) = \sqrt{1 - x^2}$$
1
vote
2answers
46 views
Proving that $\sin(a)\cos(b)$ and $\cos(a)\sin(b)$ identities are identical using $\sin(-x)=-\sin(x)$
This website states the two trig identities below are identical:
[\begin{array}{l}
\sin (a)\cos(b) = \frac{1}{2}\left[ {\sin (a + b) + \sin (a - b)} \right] \Rightarrow 1\\
\cos (a)\sin (b) = ...
1
vote
3answers
114 views
Why does tan(t) touch the unit circle at (1,0)?
I can't get my head around this, any help would be very much appreciated.
Thanks
EDIT: t is an angle, where 0 < t < 90, angle t is in degrees
EDIT: Added a picture I lifted from google
0
votes
4answers
198 views
How do I find tangent on the unit circle?
I know that $\sin=y$ and that $\cos=x$ so how do I find tangent? I've seen some diagrams where the tangent line touches the unit circle at $(1,0)$, I'm wondering how this is derived?
EDIT: ...
3
votes
2answers
242 views
Trig tricks/shortcuts
I'm just curious as to any "tricks" or shortcuts that could help making trig identities easier to solve, for example one is:
$$a\cos\theta+b\sin\theta = \sqrt{a^2+b^2}\,\cos(\theta-\phi)$$
0
votes
2answers
44 views
Area of a rectangular triangle
We need to calculate the area of the triangle shown in figure:
The text of the problem also says that: $\sin \alpha =2 \sin \beta$. What is the area of the triangle?
4
votes
7answers
335 views
Proving $\sin (x)=\cos (90^\circ-x)$
I'm interested in the different ways of proving this, any proof is welcome.
I understand one way is the cosine sum/difference formula, another is using a right angled triangle. Are there any others?
...
3
votes
7answers
112 views
How do I show $\sin^2(x+y)−\sin^2(x−y)≡\sin(2x)\sin(2y)$?
I really don't know where I'm going wrong, I use the sum to product formula but always end up far from $\sin(2x)\sin(2y)$. Any help is appreciated, thanks.
4
votes
3answers
113 views
Solving this equation $10\sin^2θ−4\sinθ−5=0$ for $0 ≤ θ<360°$
The first part of the question asks me to square both sides of the equation:
$$3 \cos θ=2 − \sin θ$$
So that I can obtain and solve the quadratic:
$$10\sin^2θ−4\sinθ−5=0 \;\;\text{for}\;\; 0 ≤ ...
3
votes
3answers
184 views
How do I find the maximum and minimum values of $1−4\cos(2\theta)+3\sin(2\theta)$?
To find the maximum of $$1 - 4\cos(2\theta) + 3\sin(2\theta) $$I tried:
$$1-4(1)+3(1)=0.$$
To find the minimum I tried to substitute with the minimum values of sin and cos:
$$1-4(-1)+3(-1)=2$$
I know ...
