Hello, everyone:
I am working on finding the exact values of angles that are less common and are therefor not found easily on the Unit Circle (at least, they are not labeled). For example, the problem I am asking about is:
10) Find the exact values of the Sine, Cosine and Tangent of 255°. We are supposed to use common angles to assist in our answers (for example, 255° = 300°-45°), and use formulas provided to solve them. I can get sine and cosine alright, but the tangent equation is causing a massive migraine:
tan(x-y) = (tan(x)-tan(y))/(1+tan(x)tan(y))
Using the formula, I get this result:
tan(300-45)=(tan(300)-tan(45))/(1+tan(300)tan(45))
Here is where I am stuck. Problem is, I did not understand the example in the notes, and the book's examples have virtually nothing to do with the actual exercise problems. So, I am trying to deduce this by logic. The denominator appears to be a conjugate, so I tried multiplying by (1-tan(300)tan(45)), and got this result:
[tan(300)+tan^2(300)tan(45)-tan(45)-tan(300)tan^2(45)]/(1-tan^2(300)tan^2(45))
Besides being nightmarishly complex, it also appears to be a dead end. I would appreciate it, greatly, if someone could take their time and slowly explain how to do this portion of my assignment?
With kind regards (except for my math teacher),
Timothy
First of all, I would not have used
225 = 300 - 45 but rather
225 = 180 + 45 and then use
tan(x+y) = (tanx + tany)/(1 - tanxtany)
you should know that tan45 = 1 and tan 180 = 0
so tan 225
= tan (180+45)
= (tan180+tan45)/(1-tan180tan45)
= 1/(1=0)
= 1
check with a calculator.
try to use combinations that involve angles like 0,30,45,60,90, 180 and 360
to use 300 would mean that you would first of all have to calculate tan 300 as a preliminary problem
Tim,
This can be done very easily by BASICS of Tangent function:
Tan has a period of PI, which is 180 degree.
When you have a angle like 225, all you need to do is add or subtract multiples of 180(in this case, 150 itself) to get a common angle:
225-180 = 45
Tan(45)=1, since 225 is in Quadrant III,
it is positive, so final answer(the whole procedure) is
tan(225)=tan(225-180)=tan(45)=1
Contact me for email help
replace 150(in the parenthesis) by 180 in my last post, that was a typo.
Hello Timothy,
I understand that finding the exact values of trigonometric functions for angles that are not commonly found on the unit circle can be challenging. Let's try to work through the problem of finding the exact values of sine, cosine, and tangent of 255° using the given formula for tangent.
First, let's break down the angle 255° into a combination of common angles:
255° = 300° - 45°
Now, let's focus on finding the exact values for sine, cosine, and tangent.
Sine (sin):
To find the sine of 255°, we can use the angle sum formula for sine:
sin(x - y) = sin(x)cos(y) - cos(x)sin(y)
In our case:
sin(255°) = sin(300° - 45°)
Now, we need to determine the values of sin(300°) and sin(45°), which are common angles.
From the unit circle:
sin(300°) = -1/2
sin(45°) = √2/2
Substituting these values into the formula:
sin(255°) = sin(300°)cos(45°) - cos(300°)sin(45°)
= (-1/2)(√2/2) - (√3/2)(√2/2)
= -√2/4 - √6/4
= (-√2 - √6)/4
So, the exact value of sine of 255° is (-√2 - √6)/4.
Cosine (cos):
To find the cosine of 255°, we can use the angle sum formula for cosine:
cos(x - y) = cos(x)cos(y) + sin(x)sin(y)
In our case:
cos(255°) = cos(300° - 45°)
Again, we need to determine the values of cos(300°) and cos(45°) from the unit circle:
cos(300°) = 1/2
cos(45°) = √2/2
Substituting these values into the formula:
cos(255°) = cos(300°)cos(45°) + sin(300°)sin(45°)
= (1/2)(√2/2) + (√3/2)(√2/2)
= √2/4 + √6/4
= (√2 + √6)/4
So, the exact value of cosine of 255° is (√2 + √6)/4.
Tangent (tan):
To find the tangent of 255°, we can use the tangent difference formula you mentioned:
tan(x - y) = (tan(x) - tan(y)) / (1 + tan(x)tan(y))
In our case:
tan(255°) = tan(300° - 45°)
We already know the values of tan(300°) and tan(45°).
From the unit circle:
tan(300°) = -√3
tan(45°) = 1
Substituting these values into the formula:
tan(255°) = (tan(300°) - tan(45°)) / (1 + tan(300°)tan(45°))
= (-√3 - 1) / (1 - √3)
To simplify further, we can multiply both the numerator and denominator by the conjugate of the denominator, which is (1 + √3):
tan(255°) = [(-√3 - 1) / (1 - √3)] * [(1 + √3) / (1 + √3)]
= [(-√3 - 1)(1 + √3)] / [(1 - √3)(1 + √3)]
= (-√3 - √3 - √9 - 1) / (1 - √9)
Simplifying:
tan(255°) = (-2√3 - 1 - 3) / (1 - 3)
= (-√3 - 4) / (-2)
= (√3 + 4) / 2
So, the exact value of tangent of 255° is (√3 + 4) / 2.
I hope this explanation helps you understand the process of finding exact values of trigonometric functions for less common angles. If you have any further questions, feel free to ask!