What is Trigonometric equation: Definition and 131 Discussions
In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
Returning if I have to show the effort, I came to this:
\frac{\sin4\alpha}{1+\cos4\alpha}\cdot\frac{\cos2\alpha}{1+\cos2\alpha}\cdot\frac{\cos\alpha}{1+\cos\alpha}=\tan\frac{\alpha}{2}.
=...
I’m stuck on how to begin. I’ve tried to factor out sin theta from both of the terms on the left hand side but that led to nowhere. Could I have a hint on how to continue? Than you!
Problem Statement : Let me copy and paste the problem as it appears in the text :
Attempt : I haven't been able to make any significant attempt at solving this problem, am afraid. I tried to reduce all the higher submultiple angles ##2\theta, 4\theta, 8\theta## into ##\theta##, but the...
Problem Statement : Solve for ##x## :
Attempt : If I take ##x=\tan\theta##, the L.H.S. reads $$\tan^{-1}\frac{1-\tan\theta}{1+\tan\theta}= \tan^{-1}\left[\tan\left(\frac{\pi}{4}-\theta \right) \right ]=\frac{\pi}{4}-\theta.$$
On going back to ##x## from ##\theta##, the given equation now...
Given : The equation ##\sin m\theta + \sin n\theta = 0##.
Attempt : Using the formula for ##\text{sin C + sin D}## (see Relevant Equation 3 above), the given equation simplifies to
\begin{equation*}
2 \sin \frac{(m+n)\theta}{2} \cos \frac{(m-n)\theta}{2} = 0
\end{equation*}
This implies the...
cot^2θ+5cosecθ=4
cot^2θ+5cosecθ-4=0
cosec^2θ+5cosec-4-1=0
cosec^2θ+5cosec-5=0
Let u=cosecθ
u^2+5u-5=0
Solve using the quadratic formula;
u=(-5± 3√5)/2
u=(-5+ 3√5)/2=0.8541...
Substitute cosecθ=u
Therefore, cosecθ=0.8541
1/sinθ=0.8541
sinθ=1/0.8541=1.170... which is not true since sin x cannot be...
a. I have just plotted the graph using desmos and attached an image here. Clearly, there are two values of x that satisfy the equation in the range. Do I need to add anything to this statement, I feel the response is a little brief for the question?
b. Using the trigonometric identities;
tan...
Summary: https://www.physicsforums.com/threads/trigonometry-question.977263/
Here's the question.
Find the solutions of the equation tan(x)=2cos(x)+1 if 0 ≤ x ≤ 2π.
I know this question can be solved by observing the graph but is there any other ways (like algorithms OR some Trigonometry...
Hi all,
I am a self learner (graduated very long ago and rusty at math) working through the Riley, Hobson and Bence text, chapter 1.
1. Homework Statement
Use the fact that ##sin(\pi/6) = 1/2## to prove that ##tan(π/12) = 2 − \sqrt{3}.##
Homework Equations
##tan(2x) = \frac { 2 tan(x)} {1...
I have a trigonometric equation
2\sin \left ( \frac{q\pi }{m} \right )-\sin \left ( \frac{q\pi }{2} \right )=0
and want to know what values m as a function of q could take to satisfy the equation. Both terms zero is the obvious solution: q=2n; m=2; n is an integer. But there are more solutions...
Homework Statement
Homework Equations
General Formula for Tan(a)=Tan(b)
The Attempt at a Solution
See the question I have uploaded.
I have tried solving it this way,
Firstly I applied the Quadratic Formula to get,
Now we have two cases,
CASE-1
When
So General Formula here will...
Homework Statement
The problem given is sin2(x) + tan2(x) = √2
2. Homework Equations
The relevant equations would be any trigonometric identities
The Attempt at a Solution
sin2(x) + tan2(x) = √2
sin2(x) + (sin2(x)/cos2(x) ) = √2
[ cos2(x) sin2(x) + sin2(x) ]/ cos2(x) = √2
[ (1- sin2(x))...
Homework Statement
##4y=cos\left(4πx+\frac{3}{2}\right)##
Homework EquationsThe Attempt at a Solution
In dividing both sides by 4, I got:
##y=\frac{1}{4}cos\left(πx+\frac{3}{8}\right)## But I am told this is incorrect.
Not sure if dividing everything by 4 here is an allowable technique, or if...
Homework Statement
##\sin a + \cos b## = ##\frac{-1}{2}##
##\cos a + \sin b## = ##\frac{\sqrt 3}{2}##
0 < a < ##\pi/2##
##\pi/2## < b < ##\pi##
a + b = ? By calculating sin (a+b)
Homework EquationsThe Attempt at a Solution
I tried :
##\sin a + \cos b =
2sin\frac{(a+b)}{2}cos\frac{(a-b)}{2}...
Homework Statement
cos2x + cos x = 0 (0 <= x <= 360)
Homework EquationsThe Attempt at a Solution
cos2x + cos x = 0
2cos(3x)/2 cos(x)/2 = 0
3x/2 = 90 degrees
x = 60 degrees
x/2 = 90
x = 180
3x/2 = 270
x = 180
x/2 = 270
x = 540 (not qualified)
is there any more possibility (answers) for x?
I've gotten to this point:
v^2/2=gs(sin(α)+cos(α)*k)
I'm suppost to get "k" from this equation can some help by showing me the steps because I'm a bit confused on how to do it.
Homework Statement
Find the solution of the inequality ## \sqrt{5-2sin(x)}\geq6sin(x)-1 ##
Answer: ## [\frac{\pi(12n-7)}{6} ,\frac{\pi(12n+1)}{6}]~~; n \in Z##Homework Equations
None.
The Attempt at a Solution
There are two cases possible;
Case-1: ##6sin(x)-1\geq0##
or...
1. Find α(β) given that the sum of the 2 sides= ##(x+y)## and its third, ##z## is a constant for 0<β<180.
You can imagine that there's two pieces of string connected between two points. One string is as long as the distance between the two points while the other string is longer. If you...
Good day :)!
Please advise how to start with the following trigonometric equation:
6*Sin^2(x) - 3*Sin^2(2x) + Cos^2(x) = 0
To be honest, I do not know what is the first steps to start with.
I have tried to start with:
5*Sin^2(x) + Sin^2(x) + Cos^2(x) - 3*Sin^2(2x) = 0
1 + 5*Sin^2(x) -...
Homework Statement :[/B]
Solve for ##x ##: $$ \sin ^{-1} {x} +\sin ^{-1} {(1-x)} =\cos ^{-1} {x} $$
Answer given: ##0## or ##\frac {1}{2}##.
Homework Equations :[/B]
All relevant formulae on inverse circular functions may be used.
The Attempt at a Solution :[/B]
Please see the pic below...
Homework Statement :[/B]
Find the general solution of the Trigonometric equation: $$3\sin ^2 {\theta} + 7\cos ^2 {\theta} =6$$
Given andwer: ##n\pi \pm \frac {\pi}{6}##
Homework Equations :[/B]
These equations may help:
The Attempt at a Solution :[/B]
Please see the pic below:
It...
Homework Statement :[/B]
Find the general solution of the equation: $$\tan {x}+\tan {2x}+\tan {3x}=0$$
Answer given: ##x=## ##\frac {n\pi}{3}##, ##n\pi \pm \alpha## where ##\tan {\alpha} = \frac {1}{\sqrt {2}}##.
Homework Equations :[/B]
These equations may be used:
The Attempt at a...
Homework Statement :[/B]
Find the general solution of the Trigonometric equation $$\sin {3x}+\sin {x}=\cos {6x}+\cos {4x} $$
Answers given are: ##(2n+1)\frac {\pi}{2}##, ##(4n+1)\frac {\pi}{14}## and ##(4n-1)\frac {\pi}{6}##.
Homework Equations :[/B]
Equations that may be used:
The...
The number of real roots of the equation
$$2cos \left( \frac {x^2 + x} {6} \right)=2^x + 2^{-x}$$
Answer options are : 0,1,2,∞
My approach :
range of cos function is [-1,1]
thus the RHS of the equation belongs to [-2,2]
So, we have
-2 ≤ 2x + 2-x ≤ 2
solving the right inequality, i got 2x...
Homework Statement
Solve acos²θ+bsinθ+c=0 for all values 0≤θ≤360°
a=16
b=6
c=-12
So 16cos²θ+6sinθ-12=0
Homework Equations
Cos²x=1-Sin²x
The Attempt at a Solution
Identity: Cos²x=1-Sin²x
16(1-Sin²θ)+6Sinθ-12=0
16-16Sin²θ+6Sinθ-12=0
6Sinθ-16Sin²θ=12-16=-4
Divide by 2(?)
3Sinθ-8Sin²θ=-2...
Homework Statement
Find all solutions of the equation in the interval [0, 2\pi].
sin6x+sin2x=0
Homework Equations
Double Angle Formulas
sin2x=2sinxcosx
cos2x=cos^{2}x-sin^{2}x
=2cos^{2}x-1
=1-2sin^{2}x
(3 formulas for cos2x)
tan2x=\dfrac{2tanx}{1-tan^{2}x}
Sum to Product Formula...
Homework Statement
\sin (x) = \frac{2}{3} and \sec (y) = \frac{5}{4}, where x and y lie between 0 and \frac{\pi}{2} evaluate \sin (x + y)
Homework Equations
Looked over some trig laws, don't think I saw anything that's too relevant. There \sec (x) = \frac{1}{\sin (x)}
The Attempt at a...
A=3sinx+4cosx and B=3cosx-4sinx if B = 4 find A.
What i tried is to use 4=3cosx-4sinx and solve for cosx
now cosx = (4+4sinx)/3 plug this into A
I end up getting A = (25sinx+16)/3 am I correct?
Homework Statement
a) Differentiate the following equation with respect to:
1) θ
2) Φ
3) ψ
(Ua - Ub)' * C * r
where:
C is a 3 x 3 rotation matrix:
[ cos θ cos ψ, -cos Φ sin ψ + sin Φ sin θ cos ψ, sin Φ sin ψ + cos Φ sin θ cos ψ]
[ cos θ sin ψ, cos Φ cos ψ + sin Φ sin θ sin...
Homework Statement
##tanx=\frac{(1+tan1)(1+tan2)-2}{(1-tan1)(1-tan2)-2}## find x
Homework Equations
3. The Attempt at a Solution [/B]
I tried multiplying through the paranthesis and arrived at ##tanx=\frac{(tan1tan2-1)+(tan2+tan1)}{(tan1tan2-1)-(tan2+tan1)}## and i don't know if this is any...
Homework Statement
Question:
Sum of all the solutions of the equation: ##tan^2 (33x) = cos(2x)-1## which lie in the interval ## [0, 314] ## is:
(a) 5050 π
(b) 4950 π
(c) 5151 π
(d) none of these
The correct answer is: (b) 4950 π
Homework Equations
## cos(2x) = 2cos^2(x) -1 ##
The Attempt...
Homework Statement
sin x = C*sin y
Find y as a function of x for a given C>0.
Homework Equations
sin x = C*sin y
The Attempt at a Solution
This is not actually a problem from a book, but a problem I myself thought about. I was studying elastic collisions in SCM and I obtained 2 equations...
http://www5a.wolframalpha.com/Calculate/MSP/MSP238521i5b83i951f19c3000010ca05be63f0bfc0?MSPStoreType=image/gif&s=10&w=219.&h=85. [Broken]
How do I solve this? I know the answers, as Wolphram Alpha has given me only the answers without any steps to how they derived those answers.
I know that...
Solve sin2x= sqrt(2)/2 (using algebra)
the interval is between 0 and 2pi
i got the first two answers: pi/8 and 3pi/8, but i don t understand how to get the other two: 9pi/8 and 11pi/8.
thanks!
Homework Statement
In 2001, Windsor, Ontario received its maximum amount of sunlight,
15.28 hrs, on June 21, and its least amount of sunlight, 9.08 hrs, on
December 21
Due to the Earth's revolution about the sun, the hours of daylight function is periodic. Determine an equation that can...
Hello reader,
I have an exam really soon and it includes a good bit of trigonometry, but I'm having problems with the trig stuff because this exam does not allow calculators and since I was dependent on the calculator, I haven't memorized anything about the trigonometric functions. I don't know...
Can anyone help with this trig question,
Determine the smallest angle in degrees such that sin theta+5cos theta=4
Iknow i need to use the quadratic formula but really stuck on it.