Without using a calculator, what is the exact value of
sec(arctan(12/5)) - sin(arccot(3/4))
5, 12, 13 right triangle here
so sec = 1/cos = 13/5
then 3, 4 , 5
tan = 1/cot = 4/3
so sin = 4/5
so
13/5 - 4/5 = 9/5
ah come on
To find the exact value of the expression sec(arctan(12/5)) - sin(arccot(3/4)), we can use trigonometric identities and the definitions of inverse trigonometric functions.
Let's start by finding the value of arctan(12/5). The arctan function gives us the angle whose tangent is equal to 12/5. We can use a right triangle to find this angle. Let's assume we have a right triangle with an angle A such that the tangent of A is 12/5. This means that the ratio of the side opposite to angle A to the side adjacent to angle A is 12/5. Let's label the side opposite to angle A as 12 and the side adjacent to angle A as 5.
Using the Pythagorean theorem, we can find the length of the hypotenuse. The hypotenuse is the side opposite the right angle. The Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides of the right triangle.
So, 12^2 + 5^2 = hypotenuse^2.
144 + 25 = hypotenuse^2.
169 = hypotenuse^2.
Taking the square root of both sides, we get
hypotenuse = 13.
Therefore, the right triangle with sides 12, 5, and 13 satisfies the conditions, and its angle A has a tangent of 12/5.
Now, we can find the value of sec(arctan(12/5)). The secant of an angle is equal to the reciprocal of the cosine of that angle. So, sec(arctan(12/5)) = 1 / cos(arctan(12/5)).
Using the right triangle we just found, we can calculate the value of cos(arctan(12/5)). We know that the cosine of an angle is equal to the ratio of the side adjacent to the angle to the hypotenuse. In our case, the side adjacent to angle A is 5, and the hypotenuse is 13.
So, cos(arctan(12/5)) = 5/13.
Therefore, sec(arctan(12/5)) = 1 / (5/13) = 13/5.
Now, let's move on to finding the value of arccot(3/4). The arccot function gives us the angle whose cotangent is equal to 3/4. We can use a right triangle to find this angle. Assuming we have a right triangle with an angle B such that the cotangent of B is 3/4, the ratio of the side adjacent to angle B to the side opposite to angle B is 3/4.
Let's label the side adjacent to angle B as 4 and the side opposite to angle B as 3.
Using the Pythagorean theorem, we can find the length of the hypotenuse. The hypotenuse is the side opposite the right angle. So, 4^2 + 3^2 = hypotenuse^2.
16 + 9 = hypotenuse^2.
25 = hypotenuse^2.
Taking the square root of both sides, we get
hypotenuse = 5.
Therefore, the right triangle with sides 3, 4, and 5 satisfies the conditions, and its angle B has a cotangent of 3/4.
Now, we can find the value of sin(arccot(3/4)). The sine of an angle is equal to the ratio of the side opposite to the angle to the hypotenuse. In our case, the side opposite to angle B is 3, and the hypotenuse is 5.
So, sin(arccot(3/4)) = 3/5.
Finally, we can substitute the values we found into the expression sec(arctan(12/5)) - sin(arccot(3/4)):
sec(arctan(12/5)) - sin(arccot(3/4)) = 13/5 - 3/5 = 10/5 = 2.
Therefore, the exact value of the expression is 2.