Angle A is in standard position and terminates in quadrant IV. If sec(A) = 4
3
, complete the steps to find
cot(A).
To find the value of cot(A) when sec(A) = 4/3, we need to use the definitions of secant and cotangent in terms of other trigonometric functions.
1. Start with the given information that sec(A) = 4/3. Recall that sec(A) is defined as the reciprocal of the cosine function: sec(A) = 1/cos(A).
2. Rearrange the equation to solve for cos(A): cos(A) = 1/sec(A) = 1/(4/3) = 3/4.
3. Now, we will use the Pythagorean Identity to find the value of sin(A) since we know cos(A). The Pythagorean Identity states: sin^2(A) + cos^2(A) = 1.
4. Substitute the value of cos(A) into the Pythagorean Identity: sin^2(A) + (3/4)^2 = 1.
Simplify the equation: sin^2(A) + 9/16 = 1.
Rearrange the equation: sin^2(A) = 1 - 9/16 = 16/16 - 9/16 = 7/16.
Take the square root of both sides to solve for sin(A): sin(A) = ±√(7/16) = ±√7/4.
Since angle A terminates in quadrant IV, where sine is negative, we take the negative square root: sin(A) = -√7/4.
5. Now, we have both sin(A) and cos(A) values. To find cot(A), we will use the definition of cotangent: cot(A) = cos(A)/sin(A).
Substitute the values of cos(A) and sin(A): cot(A) = (3/4)/(-√7/4).
Simplify the expression: cot(A) = (3/4) * (-4/√7) = -3/√7.
Therefore, cot(A) = -3/√7.
Note: The final result can also be rationalized by multiplying the numerator and denominator of cot(A) by √7: cot(A) = (-3/√7) * (√7/√7) = -3√7/7.
sec(A) = r/x, so
r = 4
x = 3
y = -√7
so, cot(A) = x/y = -3/√7