use the unit circle to investigate equivalent expressions involving the six trig functions of (x + pi) where x lies in the first quadrant. add these to your trigonometric identities table
adding π is equivalent to rotating halfway around the origin.
That changes the sign of x and y.
Now, you know that
sinθ = y/r
cosθ = x/r
tanθ = y/x
since both x and y have changed sign, and r is positive, what does that tell you about the trig functions?
that i have to use sin(pi+x)?
To investigate equivalent expressions involving the six trigonometric functions of (x + π) with x lying in the first quadrant, let's start by using the unit circle.
The unit circle is a circle of radius 1 centered at the origin (0, 0) on the Cartesian coordinate plane. It can be used to associate angles with points on the circle and determine the values of trigonometric functions for those angles.
In the first quadrant, the angle x will be between 0 and π/2. So let's focus on that range.
We'll look at each trigonometric function one by one and find the equivalent expression for (x + π):
1. Sine (sin): Sine is the y-coordinate of the point where the angle intersects the unit circle. In the first quadrant, sin x = y. Since (x + π) lies in the second quadrant, the y-coordinate will be negative. Therefore, sin(x + π) = -sin x.
2. Cosine (cos): Cosine is the x-coordinate of the point where the angle intersects the unit circle. In the first quadrant, cos x = x. Since (x + π) lies in the second quadrant, the x-coordinate will be negative. Therefore, cos(x + π) = -cos x.
3. Tangent (tan): Tangent is the ratio of sine to cosine (tan x = sin x / cos x). Therefore, tan(x + π) = sin(x + π) / cos(x + π). Using the previously determined equivalents, we have tan(x + π) = (-sin x) / (-cos x), which simplifies to tan(x + π) = tan x.
4. Cosecant (csc): Cosecant is the reciprocal of sine (csc x = 1 / sin x). Therefore, csc(x + π) = 1 / sin(x + π). Using the previously determined equivalent, we have csc(x + π) = 1 / (-sin x), which simplifies to csc(x + π) = -csc x.
5. Secant (sec): Secant is the reciprocal of cosine (sec x = 1 / cos x). Therefore, sec(x + π) = 1 / cos(x + π). Using the previously determined equivalent, we have sec(x + π) = 1 / (-cos x), which simplifies to sec(x + π) = -sec x.
6. Cotangent (cot): Cotangent is the reciprocal of tangent (cot x = 1 / tan x). Therefore, cot(x + π) = 1 / tan(x + π). Using the previously determined equivalent, we have cot(x + π) = 1 / tan x, which simplifies to cot(x + π) = cot x.
Adding these equivalent expressions to the trigonometric identities table, we have:
sin(x + π) = -sin x
cos(x + π) = -cos x
tan(x + π) = tan x
csc(x + π) = -csc x
sec(x + π) = -sec x
cot(x + π) = cot x
Now, you can incorporate these additional equivalences to enhance your trigonometric identities table.