Find the actual values for sin x by showing all your working if sec x=4/3,tan x<0.
cos² x = 1 - sin² x
cos x = ± √ ( 1 - sin² x )
sec x = 1 / cos x
sec x = 1 / ± √ ( 1 - sin² x )
4 / 3 = 1 / ± √ ( 1 - sin² x ) Take reciprocal value of both sides
3 / 4 = ± √ ( 1 - sin² x ) Raise both sides to on the power of two
9 / 16 = 1 - sin² x
1 - sin² x = 9 / 16 Subtract 1 to both sides
1 - sin² x - 1 = 9 / 16 -1
- sin² x = 9 / 16 -1
- sin² x = 9 / 16 - 16 / 16
- sin² x = - 7 / 16 Multiply both sides by - 1
sin² x = 7 / 16 Take the square root of both sides
sin x = ± √ ( 7 / 16 )
sin x = ± √7 / √16
sin x = ± √7 / 4
tan x < 0 mean tan x is negative
tan x = sin x / cos x
sec x = 4 / 3
cos x = 1 / sec x
cos x = 3 / 4
cos x is positive so:
tan x = sin x / cos x can be negative only if sin x is negative.
This mean:
sin x = - √7 / 4
if secx = 4/3
then cosx = 3/4
also since tanx < 0 we know that x must be in quadrants IV, since only in IV is the cosine positive and the tangent negative
make a sketch of a right-angled triangle in standard position with
base of 3, hypotenuse 4, and height y
x^2 + y^2 = r^2
9 + y^2 = 16
y^2 = 7
y = ± √7, but in IV y = -√7
sinx = -√7/4 , cscx = -4/√7
cosx = 3/4, secx = 4/3 <<<--- our given
tanx = -√7/3 , cotx = -3/√7
To find the actual values of sin x, given that sec x = 4/3 and tan x < 0, we can use the following steps:
Step 1: Recall the reciprocal identities:
- sin x = 1 / csc x
- cos x = 1 / sec x
- tan x = sin x / cos x
- cot x = 1 / tan x
Step 2: Given that sec x = 4/3, using the reciprocal identity cos x = 1 / sec x, we can determine the value of cos x:
- cos x = 1 / (4/3)
- cos x = 3/4
Step 3: Since tan x < 0, we know that the terminal side of angle x terminates in either the third quadrant or the fourth quadrant. In both quadrants, sin x is negative.
- sin x < 0
Step 4: Using the identity tan x = sin x / cos x and the previously found values of sin x, cos x:
- tan x = (sin x) / (cos x)
- tan x = (sin x) / (3/4)
- (sin x) / (3/4) < 0
Step 5: To satisfy the inequality (sin x) / (3/4) < 0, we can set up the following cases:
- Case 1: sin x < 0, and 3/4 > 0
- Case 2: sin x > 0, and 3/4 < 0
Step 6: In Case 1, since sin x < 0 and 3/4 > 0, there is no solution that satisfies the given conditions.
- sin x < 0 and 3/4 > 0: BONUS CASE
Step 7: In Case 2, since sin x > 0 and 3/4 < 0, we have:
- sin x > 0 and 3/4 < 0
Step 8: Now, using Pythagorean identity: sin² x + cos² x = 1, we can solve for sin x:
- sin² x = 1 - cos² x
- sin² x = 1 - (3/4)²
- sin² x = 1 - 9/16
- sin² x = 7/16
Step 9: Taking the square root of both sides, considering sin x > 0:
- sin x = √(7/16)
Step 10: Simplifying the square root:
- sin x = √7 / 4
Therefore, the actual value of sin x, given that sec x = 4/3 and tan x < 0, is √7 / 4.
To find the values of sin(x), given that sec(x) = 4/3 and tan(x) < 0, we need to follow these steps:
Step 1: Identify the relationship between sin(x), sec(x), and tan(x).
We know that sec(x) = 1/cos(x) and tan(x) = sin(x)/cos(x). By substituting these relationships into the given information, we can derive an equation to solve for sin(x).
Step 2: Substitute the given values into the equations.
We have sec(x) = 4/3 and tan(x) < 0. From the relationship between sec(x) and cos(x), we have 1/cos(x) = 4/3. Since tan(x) = sin(x)/cos(x) and tan(x) < 0, we know that sin(x) and cos(x) have opposite signs.
Step 3: Solve for cos(x).
To solve for cos(x), we can cross-multiply the equation 1/cos(x) = 4/3:
3 = 4cos(x).
Divide both sides by 4:
3/4 = cos(x).
So, we have cos(x) = 3/4.
Step 4: Determine the range of cos(x).
Since cos(x) = 3/4, we know that cos(x) is positive because it lies within the interval [0, π]. However, we also know that sin(x) and cos(x) have opposite signs, so sin(x) must be negative.
Step 5: Use the Pythagorean identity to solve for sin(x).
We can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to find sin(x). Since cos(x) = 3/4, we can substitute this value into the equation:
sin^2(x) + (3/4)^2 = 1.
sin^2(x) + 9/16 = 1.
sin^2(x) = 1 - 9/16.
sin^2(x) = 16/16 - 9/16.
sin^2(x) = 7/16.
Taking the square root of both sides gives:
sin(x) = ± √(7/16).
Since we established that sin(x) is negative, we can write:
sin(x) = - √(7/16).
Finally, simplifying further:
sin(x) = - √7/4.
Therefore, the actual value of sin(x) when sec(x) = 4/3 and tan(x) < 0 is -√7/4.