1.Write the equation 5x+y-2=0 in normal form.
5x+y-2=0
5x+y=2 (add two to both sides)
Standard form (Ax+By=C)---> 5x+y=2
Y-intercept (when x equals 0)
5x+y-2=0
5x+y=2
5(0) +y=2
y=2
X-intercept (when y equals 0)
5x+y-2=0
5x+y=2
5x+0=2
x=2/5
Slope intercept (y=mx+b)
5x+y-2=0
5x+y=2
y= -5x+2
2.Use the sum or difference identity to find the exact value of tan105 degrees.
tan 105 = tan (60 +45) = (tan 60 +tan 45)/ 1-tan60tan45 =
(sqrt 3 + 1) / (1 -sqrt3 (1)) =
(1 +sqrt 3)^2 / (1-sqrt 3)(1+sqrt 3) =
(1 +sqrt 3)^2 /( 1-3) =
(1 +2sqrt 3 + 3) / (-2) =
-2-sqrt3
#1 ok
#2 ok
To write the equation 5x+y-2=0 in normal form, we need to rearrange the equation so that all the coefficients are positive integers and the constant term is positive.
Starting with: 5x+y-2=0
We can add 2 to both sides of the equation to move the constant term to the right side:
5x+y=2
Therefore, the equation 5x+y-2=0 in normal form is 5x+y=2.
To find the y-intercept, we set x=0 and solve for y:
5x+y=2
5(0) + y = 2
y = 2
So, the y-intercept is 2.
To find the x-intercept, we set y=0 and solve for x:
5x+y=2
5x+0=2
5x=2
x = 2/5
So, the x-intercept is 2/5.
Finally, we can write the equation in slope-intercept form (y=mx+b) by solving for y:
5x+y=2
y = -5x + 2
Therefore, the equation 5x+y-2=0 in slope-intercept form is y = -5x + 2.
Now, let's move on to finding the exact value of tan 105 degrees using the sum or difference identity.
We can rewrite tan 105 degrees as tan (60 + 45) degrees.
Using the formula tan (A + B) = (tan A + tan B) / (1 - tan A tan B),
we have:
tan (60 + 45) = (tan 60 + tan 45) / (1 - tan 60 tan 45)
The exact value of tan 60 degrees is √3, and the exact value of tan 45 degrees is 1.
Plugging in these values, we have:
(tan 60 + tan 45) / (1 - tan 60 tan 45) = (√3 + 1) / (1 - √3)
To rationalize the denominator, we can multiply both numerator and denominator by the conjugate of the denominator, which is (1 + √3):
((√3 + 1) / (1 - √3)) * ((1 + √3) / (1 + √3))
Simplifying the numerator and denominator, we have:
(√3 + 1)(1 + √3) / (1 - 3)
Expanding and simplifying further:
(1 + 2√3 + 3) / (-2)
Putting it all together:
(1 + 2√3 + 3) / (-2)
= 4 + 2√3 / -2
= -2 - √3
Therefore, the exact value of tan 105 degrees is -2 - √3.