if cos ø=1/2then find the value of 2secø/1+tanø
If your question means:
2 sec ø / ( 1 + tan ø )
then
sec ø = 1 / cos ø = 1 / ( 1 / 2 ) = 2
2 sec ø = 2 ∙ 2 = 4
Cosine is positive in Quadrant I and Quadrant IV
in Quadrant I:
cos ø = 1 / 2 for ø = π / 3
tan ø = tan π / 3 = √ 3
1 + tan ø = 1 + √ 3 = √ 3 + 1
2 sec ø / ( 1 + tan ø ) = 4 / ( √ 3 + 1 )
Multiply numerator and denominator by √ 3 - 1
2 sec ø / ( 1 + tan ø ) = 4 ( √ 3 - 1 ) / [ ( √ 3 + 1 ) ( √ 3 - 1 ) ]
Since ( a + b ) ( a - b ) = a² - b²
2 sec ø / ( 1 + tan ø ) = 4 ( √ 3 - 1 ) / [ ( √ 3 )² - 1² ]
2 sec ø / ( 1 + tan ø ) = 4 ( √ 3 - 1 ) / ( 3 - 1 )
2 sec ø / ( 1 + tan ø ) = 4 ( √ 3 - 1 ) / 2
2 sec ø / ( 1 + tan ø ) = 2 ( √ 3 - 1 )
in Quadrant IV:
cos ø = 1 / 2 for ø = 5 π / 3
tan ø = tan 5 π / 3 = - √ 3
1 + tan ø = 1 - √ 3
2 sec ø / ( 1 + tan ø ) = 4 / ( 1 - √ 3 )
Multiply numerator and denominator by 1 + √ 3
2 sec ø / ( 1 + tan ø ) = 4 ( 1 + √ 3 ) / [ ( 1 - √ 3 ) ( 1 + √ 3 ) ]
Since ( a - b ) ( a + b ) = a² - b²
2 sec ø / ( 1 + tan ø ) = 4 ( 1 + √ 3 ) / [ 1² - ( √ 3 )² ]
2 sec ø / ( 1 + tan ø ) = 4 ( √ 3 + 1 ) / ( 1 - 3 )
2 sec ø / ( 1 + tan ø ) = 4 ( √ 3 + 1 ) / ( - 2 )
2 sec ø / ( 1 + tan ø ) = - 2 ( √ 3 + 1 )
Without finding the actual angle:
if cosø = 1/2, then secø = 2
x^2 + y^2 = r^2
1 + y^2 = 4
y = ± √3 , and sinø = ± √3/2,
and tanø = sinø/cosø
= ±(√3/2)/(1/2)
= ±√3
In quad I:
2secø/(1+tanø) = 4/(1 + (√3/2)/(1/2)) = 4/(1+√3)
in quad IV:
2secø/(1+tanø) = 4/(1 - (√3/2)/(1/2)) = 4/(1-√3)
To find the value of 2secø / (1 + tanø) given that cosø = 1/2, we can use the trigonometric identities:
secø = 1 / cosø
tanø = sinø / cosø
First, let's find the value of 2secø.
Since secø = 1 / cosø, we can substitute the value of cosø into the equation:
secø = 1 / (1/2) = 2
Now, let's find the value of tanø.
Since tanø = sinø / cosø, we need to find the value of sinø.
Using the identity sin^2ø + cos^2ø = 1, we can substitute the value of cosø:
sin^2ø + (1/2)^2 = 1
sin^2ø + 1/4 = 1
sin^2ø = 1 - 1/4
sin^2ø = 3/4
Taking the square root of both sides, we get:
sinø = √(3/4)
sinø = √3 / 2
Now, substitute the values of sinø and cosø into the equation for tanø:
tanø = (sinø / cosø) = (√3 / 2) / (1/2) = √3
Finally, substitute the values of secø and tanø into the expression:
2secø / (1 + tanø) = 2 / (1 + √3)
Therefore, the value of 2secø / (1 + tanø) when cosø = 1/2 is 2 / (1 + √3).