If a is an angle in standard position and its terminal side passes through the point (-3,2), find the exact value of csc a.
Answer: radical 13/2
A large pizza has a diameter of 16 inches. Which of the following is the area of the pizza in terms of pi (n) (a) 32 (b) 64 (c) 13 (d) 10
To find the exact value of csc(a), we need to determine the quadrant in which the angle a lies.
Given that the terminal side of angle a passes through the point (-3,2), we can determine the position by considering the signs of the x and y coordinates.
Since the x-coordinate (-3) is negative and the y-coordinate (2) is positive, we can conclude that the point (-3,2) lies in the second quadrant.
In the second quadrant, the values of the trigonometric functions take on specific signs:
- sin(a) is positive,
- cos(a) is negative, and
- tan(a) is positive.
Since csc(a) is the reciprocal of sin(a), csc(a) will also be positive.
To find the exact value of csc(a), we need to find the value of sin(a).
Using the Pythagorean theorem, we can determine the hypotenuse of the right triangle formed by the point (-3,2):
h = √((-3)^2 + 2^2)
= √(9 + 4)
= √13
Now, we can find sin(a) using the y-coordinate and the hypotenuse:
sin(a) = opposite/hypotenuse
= 2/√13
Finally, the exact value of csc(a) is the reciprocal of sin(a):
csc(a) = 1/sin(a)
= 1/(2/√13)
= √13/2
Therefore, the exact value of csc(a) is √13/2.
To find the exact value of csc(a), we need to determine the sine of angle a first.
Given that the terminal side of angle a passes through the point (-3,2), we can use the Pythagorean theorem to find the length of the hypotenuse (r). The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (r^2) is equal to the sum of the squares of the other two sides.
In this case, the hypotenuse is the distance from the origin (0,0) to the point (-3,2), which is r. So, using the coordinates of the point, we have:
r^2 = (-3)^2 + 2^2
= 9 + 4
= 13
Since we know the hypotenuse, we can find the sine of angle a by dividing the length of the opposite side by the hypotenuse. In this case, the opposite side is the y-coordinate of the point (-3,2), which is 2.
So, sin(a) = opposite/hypotenuse
= 2/√13
Now, to find the csc(a), we take the reciprocal of the sine of angle a:
csc(a) = 1/sin(a)
= 1 / (2/√13)
= √13/2
Therefore, the exact value of csc(a) is √13/2.
draw the triangle.
Csc= hypotenuse/opposite= sqrt((-3)^2+2^2) /2
csca= 1/2 sqrt13