Let an angle in quad IV such that cos= 2/5 find exact value of csc and cot
you have a 2-√21-5 triangle, so in QIV, you have
cos = 2/5
csc = -5/√21
cot = cos*csc = -2/√21
To find the exact values of csc and cot given that cos = 2/5, we need to use the definitions and properties of trigonometric functions. Let's go step by step.
First, we know that cos = adjacent/hypotenuse. In quadrant IV, the adjacent side is positive, and the hypotenuse is positive. Let's assume the adjacent side is 2 and the hypotenuse is 5 (since cos = 2/5).
Using the Pythagorean theorem, we can find the length of the opposite side (let's call it b):
b² = (hypotenuse)² - (adjacent)²
b² = 5² - 2²
b² = 25 - 4
b² = 21
b = √21 (Since the length of a side cannot be negative)
Now that we know the opposite side length is √21, we can find the values of csc and cot.
1. Cosecant (csc) can be found using the reciprocal of the sine function:
csc = 1/sin
Since sin = opposite/hypotenuse, we have sin = (√21)/5
Therefore, csc = 1/(√21/5) = 5/√21
To rationalize the denominator, we multiply both numerator and denominator by √21:
csc = (5/√21) * (√21/√21) = 5√21/21
So, the exact value of csc is 5√21/21.
2. Cotangent (cot) can be found using the reciprocal of the tangent function:
cot = 1/tan
Since tan = opposite/adjacent, we have tan = (√21)/2
Therefore, cot = 1/(√21/2) = 2/√21
To rationalize the denominator, we multiply both numerator and denominator by √21:
cot = (2/√21) * (√21/√21) = 2√21/21
So, the exact value of cot is 2√21/21.
To summarize:
- The exact value of csc is 5√21/21.
- The exact value of cot is 2√21/21.