# General solution of trig functions

• t_n_p
In summary, the general solutions for the given trigonometric equations are x = (π/4) + 2nπ and x = π/3 + 2nπ for a) and x = 2nπ ± acos(0.5) for b). The second general solution for a) is x = (2n+1)π - asin(1/\sqrt{2}) and for b) it is x = 2nπ ± acos(0.5). The methods for obtaining these solutions involve sketching the graphs of sine and cosine and using the unit circle to understand the symmetry of the functions. The ± in the solutions accounts for the different quadrants in which the functions are positive.
t_n_p

## Homework Statement

Find the general solution of:
a) sin x = 1/$$\sqrt{2}$$
b) cos x = 0.5

## The Attempt at a Solution

a) x = asin(1/$$\sqrt{2}$$)
x = (π/4) + 2nπ

b) x = acos(0.5)
x = π/3 + 2nπ

Basically, my strategy was to solve for the basic angle, and then add multiples of the period (2nπ for sin and cos, nπ for tan).

a) x = 2nπ + asin(1/$$\sqrt{2}$$)
or x = (2n+1)π - asin(1/$$\sqrt{2}$$)

b) 2nπ ± acos(1/2)

Just wondering,

for a) how do they find the second solution x = (2n+1)π - asin(1/$$\sqrt{2}$$)? I'm curious to know, as the follow on question asks for the first 2 positive solutions to sin x = 1/$$\sqrt{2}$$, which can apparently only be obtained using both answers shown above. I only know how to get the first solution

for b) where does the ± come from? My answer is the same but for ±.

hmm interesting.
got an email notification saying there was a reply to this thread, but there is nothing here!

guess posted must have deleted!

Have you sketched both of them and tried to reason by looking at the sketches? It should be quite helpful to do that.

Some things to note; cos is an even function as you'll see when you sketch it which is related to the +-.

Sin is an odd function, and therefore is unlikely to have just one solution for all, hopefully when you sketch it you'll see that.

I understand graphically how the solutions are valid, but is there a set method/recipe to obtain these results?

Would it be right ro assume, that for the cos question, my method is correct, it just needs the plus/minus in there?

as for the sin question, how they get the second general solution is still a puzzle to me...

Just adding multiples of 2pi is not enough (saying that sin(x) has period 2pi means that if y and x differ by a multiple of 2pi, then sin(x)= sin(y). It does NOT say that if sin(x)= sin(y) then they must differ by a multiple of 2pi.

One way to remember the connections is to think of sine and cosine in terms of the unit circle. If (x, y) is a point on the unit circle so that the radius make angle $\theta$ with the positive x-axis, then $x= cos(\theta)$ and $y= sin(\theta)$. That means that 2 angles, $\theta$ and $\phi$, between 0 and $2\pi$ will have the same sine if they lie on same horizontal line (same y value) and such a line will intersect the unit circle twice: by symmetry, $\phi= \pi- \theta$. Similarly with cosine and a vertical line: $cos(\theta)= cos(2\pi- \theta)$.

ok yep, definitely see what you are saying (halls of ivy).

when I look at sin, positive in 1st and 2nd quadrants, then when I look at the solutions it makes sense. same with cos, positive in 1st and 4th quadrants, the plus minus accounts for this.

Naturally this leads to the following question:

find general solution of:
tanx = -3/$$\sqrt{3}$$
x = atan(-3/$$\sqrt{3}$$)

tan negative in 2nd and fourth quadrants, so general equation(s) are
(2n-1)π - atan(-3/$$\sqrt{3}$$)
2nπ - atan(-3/$$\sqrt{3}$$)

Sound good?

bump.

the negative in the atan(...) is bothering me.
I still need to be convinced!

bump
still seeking clarification with the negative case

## 1. What is a general solution of trig functions?

A general solution of trig functions is an equation that can be used to find all possible solutions to a trigonometric equation. It includes a constant or variable that allows for a range of solutions to be obtained.

## 2. How is a general solution of trig functions different from a specific solution?

A specific solution of a trig function refers to a single solution that satisfies a given trigonometric equation. A general solution, on the other hand, includes a constant or variable that allows for multiple solutions to be obtained.

## 3. Can a general solution of trig functions be used to find all possible solutions to a trigonometric equation?

Yes, a general solution can be used to find all possible solutions to a trigonometric equation. By including a constant or variable, the general solution allows for a range of solutions to be obtained.

## 4. What are some common examples of general solutions of trig functions?

Examples of general solutions of trig functions include sin(x) = k, cos(x) = k, and tan(x) = k, where k is a constant or variable. These equations can be rearranged to solve for x and obtain multiple solutions.

## 5. How can a general solution of trig functions be used in real-world applications?

General solutions of trig functions can be used in a variety of real-world applications, such as in engineering, physics, and navigation. For example, the general solution sin(x) = k can be used to find the possible angles of elevation or depression in a structure or the trajectory of a projectile.

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