Trigonometry: What does this mean and how do you solve it???

Question

Hi! This is in the section of my trigonometry book on trigonometric identities.... specifically the sum and difference formulas for sin and tan functions. Anyway, the directions say to "Write each expression as a function of α alone." I have no idea what that means or how to do it.

Here's a few examples:

sin (α - π)
sin (360° - α)
tan (π/4 + α)
tan (180 + α)

What does that mean and how do you do that to problems like the above? Thank you so much! I'll pick a best answer TODAY!

Thank you! I know the answers though (I have the answer key in the back of the book), so I was hoping that you'd explain how exactly you got those answers if you could. :)

Answer 1

I would say it's a poorly phrased question, since each of those is a function of α alone, technically. But, what they mean is to write each expression so that only α appears in the trig functions. For instance:
sin( A + B) = sin(A)cos(B) + sin(B)cos(A)
so
sin (α - π) = sin(α) cos(-π) + sin(-π) cos(α)
cos(-π) = -1 sin(-π) = 0
Therefore:
sin (α - π) = -sin(α)
Basically, just use the formulas for the trig sums to get the numbers out of the trig functions.

Answer 2

the idea here is just to use the degrees or radians to add a horizontal shift in the functions.

sin (α - π) = - sin( α ) as the π radians shifts the sine function half way around the unit circle.

sin (360° - α) = sin( - α ) = - sin (α )

tan (π/4 + α) = (cos( α ) + sin( α )) / (cos( α ) - sin( α ))
from tan(A+B) = (tan(A) + tan(B)) / (1 -tan(A)tan(B))

tan (180 + α) = -1 / tan( α )

look at the unit circle and see how the functions move around based on the horizontal shift.

Answer 3

tan (x +y) = (tan x +tan y)/(1-tanxtany)
So tan (π/4 + α)=(tanα +tan(π/4))/(1 -tanαtan(π/4))
But tan(π/4) = 1
So we have (π/4 + α)=(tanα +1)/(1-tanα)

The other problems would be done exactly the same way.

Answer 4

sin (α - π) = -sinα, since sin (α - π) = -sin (π-α)

sin (360° - α) = -sinα, since sin (360° - α) = sin (-α)

tan (π/4 + α) = (1+tanα)/(1-tanα), since tan(x+y) = (tanx + tany)/(1-tanx*tany), and tan(π/4) = 1.

tan (180 + α) = tanα, since 180 is the period of tanx.

Answer 5

Draw a photograph of a ladder leaning up against a house. then you certainly could desire to ascertain that the floor and the living house make a suited attitude, you comprehend the dimensions of the hypotnuse (40 two ft), and you comprehend how severe up this is (25 ft) so for you to use the an inverse sin function (theata)=sin^-a million(25/40 two)

Answer 6

just follow the formula on your book

sin (a-b) = sin (a+ (-b))

for sin (α - π) :

sin (α - π) = sin α cos (-π) + sin (-π) cos α

note:

cos -π = -1 and

sin π = 0

so:

sin (α - π) = -sin α


please correct me if im wrong..