Let θ be the angle between the x-axis and the line connecting the origin O(0,0) and the point P(−3,−4), where 180∘<θ<270∘. Given that sinθ+cosθ+tanθ=−a/b, where a and b are coprime positive integers. What is the value of a+b?
Well, well, well! It seems like we've got ourselves a trigonometry problem with a little bit of humor thrown in, just for fun! Let's dive right in, shall we?
First things first, let's find the value of sinθ, cosθ, and tanθ. We have the point P(-3, -4), which is a classic "3-4-5" right triangle! So, sinθ = -4/5, cosθ = -3/5, and tanθ = -4/-3 = 4/3.
Now, we need to find the value of a/b. Adding up sinθ, cosθ, and tanθ, we get:
sinθ + cosθ + tanθ = -4/5 - 3/5 + 4/3 = -12/15 - 9/15 + 20/15 = -1/15.
So, our answer is -1/15, which means a = -1 and b = 15. Since we want the sum of a and b, we simply calculate a + b:
-1 + 15 = 14.
And there you have it, my friend! The value of a + b is 14. Keep that smile on your face and those math problems coming!
To find the value of a and b, we need to find the values of sinθ, cosθ, and tanθ.
1. Start by finding the length of the line connecting the origin O(0,0) and the point P(-3,-4) using the distance formula:
d = √[(-3 - 0)² + (-4 - 0)²]
= √[(-3)² + (-4)²]
= √[9 + 16]
= √25
= 5
2. Since the angle θ is between the x-axis and the line OP, we can calculate sinθ and cosθ using the coordinates of point P:
sinθ = y-coordinate of P / length of OP
= -4 / 5
= -4/5
cosθ = x-coordinate of P / length of OP
= -3 / 5
= -3/5
3. We can find tanθ by using the trigonometric identity tanθ = sinθ / cosθ:
tanθ = (-4/5) / (-3/5)
= 4/3
4. Now, we can substitute these values into the equation sinθ + cosθ + tanθ = -a/b:
(-4/5) + (-3/5) + (4/3) = -a/b
Simplifying the equation further gives: (-12 + -9 + 20) / (15) = -a/b
(-1) / (15) = -a/b
Since a and b are positive integers, we multiply both sides of the equation by -15:
-15 * (-1/15) = -a/b * -15
1 = a/b
Therefore, a = 1 and b = 1.
5. Finally, the value of a + b = 1 + 1 = 2.
So, the value of a + b is 2.
To find the value of a+b, we need to determine the values of sinθ, cosθ, and tanθ using the given information about the angle θ.
Let's start by finding the lengths of the sides of the right triangle formed by the line connecting the origin O(0,0) and point P(−3,−4). Using the Pythagorean theorem, we can find the length of the line segment OP, which is the hypotenuse of the triangle:
OP = √((-3)^2 + (-4)^2)
= √(9 + 16)
= √25
= 5
Now, we can find the values of sinθ and cosθ:
sinθ = opposite/hypotenuse = -4/5
cosθ = adjacent/hypotenuse = -3/5
Next, we can find the value of tanθ using the formula:
tanθ = sinθ/cosθ = (-4/5)/(-3/5)
= 4/3
Finally, we can substitute the values of sinθ, cosθ, and tanθ into the equation sinθ + cosθ + tanθ = -a/b to find the value of a and b:
-4/5 - 3/5 + 4/3 = -a/b
(-12 - 9 + 20)/(15) = -a/b
(-1)/(15) = -a/b
From here, we can see that a = 1 and b = 15, as -1 and 15 are coprime positive integers.
Therefore, the value of a+b is 1 + 15 = 16.