determine the exact value of (cot pi/3) + (csc pi/3) / (cos pi/4). how is this calculated? i mean the steps to get the exact value?
all those angles are the standard ones you should know.
cot pi/3 = 1/√3
csc pi/3 = 2/√3
cos pi/4 = 1/√2
plug it in and you have
(1/√3 + 2/√3)/(1/√2)
= √6
To determine the exact value of the expression (cot π/3) + (csc π/3) / (cos π/4), we need to use trigonometric identities and apply the values of trigonometric functions for specific angles.
Step 1: Identify the given angles
π/3 and π/4 are the angles given in the expression.
Step 2: Determine the values of trigonometric functions
Recall the values of trigonometric functions for specific angles:
- sin(θ) = 1 / csc(θ)
- cos(θ) = 1 / sec(θ)
- tan(θ) = 1 / cot(θ)
Step 3: Calculate the values of each trigonometric function for the given angles
For π/3:
- cos(π/3) = 1 / 2
- sin(π/3) = √3 / 2
- tan(π/3) = √3
For π/4:
- cos(π/4) = 1 / √2
- sin(π/4) = 1 / √2
- tan(π/4) = 1
Step 4: Substitute the values into the expression
Substituting the known values, the expression becomes:
(cot π/3) + (csc π/3) / (cos π/4)
= (1 / tan π/3) + (1 / sin π/3) / (cos π/4)
= (1 / √3) + (2 / √3) / (1 / √2)
Step 5: Simplify the expression
Now, let's simplify the expression by rationalizing the denominator:
((1 / √3) + (2 / √3)) * (√2 / √2)
= ((1 * √2) + (2 * √2)) / (√3 * √2)
= (√2 + 2√2) / √6
= 3√2 / √6
Step 6: Rationalize the denominator
To rationalize the denominator, multiply the fraction by (√6 / √6):
(3√2 / √6) * (√6 / √6)
= (3√12) / 6
= √12 / 2
Step 7: Simplify the expression
Finally, simplify the expression:
√12 / 2 = (√4 * √3) / 2
= (2 * √3) / 2
= √3
Therefore, the exact value of (cot π/3) + (csc π/3) / (cos π/4) is √3.
To determine the exact value of the expression (cot π/3) + (csc π/3) / (cos π/4), we need to understand the trigonometric functions involved and the relationships between them.
Before we proceed, let's define the trigonometric functions for clarity:
- cot(θ) = 1/tan(θ) = cos(θ) / sin(θ)
- csc(θ) = 1/sin(θ)
- cos(θ) = adjacent/hypotenuse
Now, let's evaluate the expression step by step:
Step 1: Find the value of cot(π/3)
cot(π/3) is equivalent to cos(π/3) / sin(π/3). We know that π/3 corresponds to an angle of 60 degrees, and for a 60-degree angle in a right-angled triangle, the adjacent side is 1, the opposite side is √3, and the hypotenuse is 2.
cos(π/3) = adjacent/hypotenuse = 1/2
sin(π/3) = opposite/hypotenuse = √3/2
Therefore, cot(π/3) = cos(π/3) / sin(π/3) = (1/2) / (√3/2) = 1/√3.
Step 2: Find the value of csc(π/3)
csc(π/3) is equivalent to 1/sin(π/3). Using the values mentioned above, we have:
sin(π/3) = opposite/hypotenuse = √3/2
Therefore, csc(π/3) = 1/sin(π/3) = 1 / (√3/2) = 2/√3.
Step 3: Find the value of cos(π/4)
cos(π/4) corresponds to a 45-degree angle. In a right-angled triangle, both the adjacent and opposite sides are equal because it is an isosceles triangle. Therefore:
cos(π/4) = adjacent/hypotenuse = opposite/hypotenuse = 1/√2.
Step 4: Substitute the calculated values into the expression
Now that we have all the values, we can substitute them into the expression:
(cot(π/3) + csc(π/3)) / cos(π/4) = (1/√3 + 2/√3) / (1/√2).
Step 5: Simplify the expression
To simplify, we need to rationalize the denominators, which involves multiplying both the numerator and denominator by the conjugate of the denominator, in this case, √2.
((1/√3 + 2/√3) / (1/√2)) * (√2/√2)
= ((1/√3 + 2/√3) * √2) / 1
= (√2/√3 + 2√2/√3) / 1
= (√2 + 2√2) / √3
= (3√2) / √3
= (3√2 * √3) / (√3 * √3)
= (3√6) / (√9)
= 3√6 / 3
= √6.
Therefore, the exact value of the given expression is √6.