Let θ be an angle in quadrant IV such that cosθ = 4/9.
Find the exact values of csc θ and cot θ.
I am just completely lost! Thanks!
4 along x axis
9 along hypotenuse sloping down
so y = - sqrt(81 -16) = -sqrt(65)
so sin T = -sqrt(65) /9
so csc T = 1/sin T = -9/sqrt 65
cot T = cos T/sin T = (4/9 ) / -(sqrt 65)/9
= 4/sqrt 65 = (4 sqrt 65) / 65
Well, you're in luck, because I'm here to make things a little less serious! Let's see if I can cheer you up while helping you with your math problem.
First, we're given that cosθ = 4/9. In quadrant IV, only sine, cosecant, and cotangent are positive. So, we can start by finding sinθ using the Pythagorean identity.
Since cosθ = 4/9, we can find sinθ using the identity: sin^2θ + cos^2θ = 1.
Plugging in the values, we have: sin^2θ + (4/9)^2 = 1.
To simplify, let's find a common denominator: sin^2θ + 16/81 = 1.
Rearranging the equation, we get: sin^2θ = 1 - 16/81.
Combining the numbers, we have: sin^2θ = 81/81 - 16/81.
Simplifying further, sin^2θ = 65/81.
Taking the square root of both sides, we get: sinθ = ±√(65/81).
Since we're in quadrant IV, where sine is positive, we can conclude that sinθ = √(65/81).
Now, let's move on to finding cscθ and cotθ.
Cosecant (csc) is the reciprocal of sine, so cscθ = 1/sinθ.
Applying this, we have: cscθ = 1/√(65/81).
To simplify further, let's rationalize the denominator: cscθ = √(81/65) / (65/81).
Dividing the fractions, we get: cscθ = √81*81 / √65*65.
Simplifying, cscθ = 81/65.
Finally, let's find cotθ. Cotangent (cot) is the reciprocal of tangent, and tangent is the ratio of sine to cosine: cotθ = cosθ / sinθ.
Plugging in the values, cotθ = (4/9) / √(65/81).
To simplify, we'll multiply the numerator and denominator by the conjugate of the denominator: cotθ = (4/9) * (√81/√65).
Further simplifying, cotθ = (4/9) * (9/√65).
Multiplying the fractions, cotθ = 4/√65.
So, in conclusion:
cscθ = 81/65
cotθ = 4/√65.
Hope that brought a smile to your face! Keep up the good work!
To find the exact values of csc θ and cot θ, we can use the given value of cos θ = 4/9 and use the relationships between trigonometric functions in a right triangle.
Since θ is in quadrant IV, we know that the adjacent side will be positive and the opposite side will be negative.
Let's start by finding the sine of θ using the Pythagorean identity:
sin^2 θ + cos^2 θ = 1
Substitute the given value of cos θ:
sin^2 θ + (4/9)^2 = 1
sin^2 θ + 16/81 = 1
sin^2 θ = 1 - 16/81
sin^2 θ = 65/81
Taking the square root of both sides gives us:
sin θ = ± √(65/81)
Since θ is in quadrant IV, the sine will be negative:
sin θ = -√(65/81)
Now, we can use the reciprocal identities to find the values of csc θ and cot θ.
csc θ = 1/sin θ
csc θ = 1/(-√(65/81))
csc θ = -√(81/65)
csc θ = -√(81/65) * (√(65)/√(65))
csc θ = -√(81*65)/(√(65)*√(65))
csc θ = -9√65/65
cot θ = cos θ/sin θ
cot θ = (4/9)/(-√(65/81))
cot θ = (4/9)/(-√(65)/√(81))
cot θ = (4/9)*(-√81/√65)
cot θ = (-4/9)*(9/√65)
cot θ = -4√65/65
So, the exact values of csc θ and cot θ are:
csc θ = -9√65/65
cot θ = -4√65/65
To find the exact values of csc θ and cot θ, we'll need to use the information given about cosθ.
Step 1: Determine the value of sin θ
Since cos θ is positive in the fourth quadrant, we know that sin θ is negative. Using the Pythagorean identity sin²θ + cos²θ = 1, we can solve for sin θ:
sin²θ = 1 - cos²θ
sin²θ = 1 - (4/9) = (9/9) - (4/9) = 5/9
Taking the square root of both sides, we get:
sin θ = -√(5/9) = -√5/3
Step 2: Find the value of csc θ
The cosecant function (csc) is equal to the reciprocal of the sine function. Therefore, csc θ = 1/sin θ. Substituting the value of sin θ, we have:
csc θ = 1/(-√5/3)
csc θ = -3/√5
To rationalize the denominator, we multiply the numerator and denominator by √5:
csc θ = -3/√5 * √5/√5
csc θ = -3√5/5
So, csc θ = -3√5/5.
Step 3: Determine the value of cot θ
The cotangent function (cot) is equal to the reciprocal of the tangent function. Since we know that cot θ = cos θ/sin θ, we can substitute the values of cosθ and sinθ:
cot θ = cos θ / sin θ
cot θ = (4/9) / (-√5/3)
To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction:
cot θ = (4/9) * (3/-√5)
cot θ = -12/9√5
Next, we can simplify the fraction by dividing both the numerator and denominator by the greatest common divisor, which is 3:
cot θ = -4/3√5
So, cot θ = -4/3√5.
Therefore, the exact values of csc θ and cot θ are -3√5/5 and -4/3√5, respectively.