I am really good at finding general solutions of the trigonometric equation by looking at the graph and by using CAST rule,but when I looked at some books there were some formulas of sin,cos and tan which will give all the solutions of the given equation which I couldnot understand.Please help me through this with the formulas to find all the solutions of sin,cos and tan.
Allright let's have an equation tan theta=1
We can have the general solution of theta as 45 degrees and 180+45=225 degrees.But I want to find other solutions of theta by using formula,so is there any specific formula of finding solutions of tan,sin and cos.[/COLOR]
Since there is no restriction on , we can attempt to generalize this.
Since when and , we can see that we can have a radian difference between the angles [either postive or negative].
So we can have
This can be generalized as
Does this process make sense?
You want to find the sin, cos, tan, cot, sec and csc of theta from just the given
tan(theta) = 1.
You want it done by formulas. Not by the reference right triangle of theta.
Then we will go to the Pythagorean trig identity because from there we can establish relationships between two trig functions in one equation.
sin^2(theta) +cos^2(theta) = 1 -----the Pythagorean trig identity.
Since the given equation, tan(theta) = 1, in in tan only, then we will use the Pythagorean trig identy where tan is involved.
Divide both sides of the said identity by cos^2(theta),
tan^2(theta) +1 = sec^2(theta)
1^2 +1 = sec^2(theta)
sec^2(theta) = 2
sec(theta) = sqrt(2) ----------answer.
cos(theta) = 1/sec(theta) = 1/sqrt(2) -------answer.
tan(theta) = sin(theta) / cos(theta)
sin(theta) = tan(theta) *cos(theta) = 1 * 1/sqrt(2) = 1/sqrt(2) ------answer.
csc(theta) = 1/sin(theta) = 1 / (1/sqrt(2)) = sqrt(2) -----answer.
cot(theta) = 1/tan(theta) = 1/1 = 1 --------answer.
Those are all for theta being in the first quadrant only. If you want to solve for the different trig functions values when theta is in the 3rd quadrant also, then just be carefull with th + and - signs.