sec(cos-1 (4/5))=?
(cos-1 (4/5))
means : find the angle Ø so that cos Ø = 4/5
so clearly secØ = 5/4
To calculate sec(cos^-1(4/5)), we'll follow these steps:
Step 1: Start with the given angle of cos^-1(4/5).
Step 2: Use the definition of cosine to find the corresponding side lengths of a right triangle.
Step 3: Use the Pythagorean theorem to find the remaining side length.
Step 4: Finally, use the definition of secant to find sec(cos^-1(4/5)).
Let's dive into each step in detail:
Step 1: Start with the given angle of cos^-1(4/5).
The expression cos^-1(4/5) represents the inverse cosine of 4/5. This means cos(cos^-1(4/5)) should equal 4/5.
Step 2: Use the definition of cosine to find the corresponding side lengths of a right triangle.
Since cos(angle) = adjacent/hypotenuse, we have cos(cos^-1(4/5)) = adjacent/hypotenuse = 4/5.
This implies that the adjacent side of the right triangle is 4 and the hypotenuse is 5.
Step 3: Use the Pythagorean theorem to find the remaining side length.
Using the Pythagorean theorem, we can find the length of the opposite side of the right triangle.
a² + b² = c²
where a and b are the two legs of the triangle and c is the hypotenuse.
In this case, a = 4 and c = 5. Let's solve for b:
4² + b² = 5²
16 + b² = 25
b² = 25 - 16
b² = 9
b = √(9)
b = 3
So, the length of the opposite side is 3.
Step 4: Finally, use the definition of secant to find sec(cos^-1(4/5)).
The definition of secant is sec(angle) = hypotenuse/adjacent.
In this case, sec(cos^-1(4/5)) = hypotenuse/adjacent = 5/4.
Therefore, sec(cos^-1(4/5)) = 5/4.
To find the value of sec(cos^(-1)(4/5)), we can use the trigonometric identity:
sec(x) = 1 / cos(x)
In this case, we need to find the value of cos^(-1)(4/5) first.
To do that, we use the inverse cosine function (also known as arccosine or cos^(-1)). This function returns an angle whose cosine is equal to the given value.
So, cos^(-1)(4/5) means finding an angle whose cosine is 4/5.
Using a calculator or a trigonometric table, we find that the angle whose cosine is 4/5 is approximately 36.87 degrees (or π/5 radians).
Now, we can substitute this angle into the sec(x) formula:
sec(cos^(-1)(4/5)) = sec(36.87 degrees)
Finally, we can evaluate sec(36.87 degrees) using a calculator to find the exact value.