how do you find the function csc 225 degrees without using a calculator?
It will have the same sign as 1/csc225 = sin 225, which is negative in the third quadrant.
The absolute value is the same as that of reference angle to the 180 degree, -x axis, which is 45 degrees.
csc 45 = 1/sin 45 = sqrt 2
The value of csc 225 is therefore
-sqrt2 = -1.414
thank you drwls!
To find the value of the csc(225 degrees) without using a calculator, you can use the concept of reciprocal trigonometric functions and trigonometric identities.
Step 1: Convert the angle to its equivalent angle in the first quadrant.
- Since 225 degrees is in the third quadrant, we can find an equivalent angle in the first quadrant by subtracting 180 degrees from it.
- 225 - 180 = 45 degrees
Step 2: Identify the reference angle.
- In the first quadrant, the reference angle is the angle between the terminal side and the x-axis.
- In this case, the reference angle is 45 degrees.
Step 3: Identify the related acute angle.
- The related acute angle is the complement of the reference angle.
- In this case, the related acute angle is 90 - 45 = 45 degrees.
Step 4: Use the trigonometric identity.
- The reciprocal identity for csc is csc(x) = 1/sin(x).
- Since the related acute angle of 45 degrees contains a right triangle, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to solve for sin(45 degrees) and csc(45 degrees).
- In this case, sin(45 degrees) = cos(45 degrees) = sqrt(2) / 2.
Step 5: Calculate the csc(45 degrees).
- csc(45 degrees) = 1 / sin(45 degrees) = 1 / (sqrt(2) / 2) = 2 / sqrt(2) = sqrt(2).
Therefore, the value of csc(225 degrees) without using a calculator is equal to sqrt(2).
To find the value of the csc (cosecant) function of 225 degrees without using a calculator, you can follow these steps:
1. Recall the relationship between csc θ and sin θ: csc θ = 1/sin θ.
2. Determine the value of sin 225 degrees. To do this, remember that sin θ is positive in the third quadrant, where sine is represented by the y-coordinate in the unit circle.
3. Use the symmetry property of the unit circle to find the reference angle: 225 degrees - 180 degrees = 45 degrees.
4. Calculate the sine of the reference angle: sin 45 degrees = √2/2.
5. Remember that in the third quadrant, the y-coordinate is negative, so sin 225 degrees = -√2/2.
6. Find the reciprocal of sin 225 degrees to get the value of csc 225 degrees: csc 225 degrees = 1/(-√2/2) = -2/√2.
7. Simplify the denominator by rationalizing it: multiplying the numerator and denominator by √2 gives csc 225 degrees = -2√2/2 = -√2.
Therefore, the value of the csc 225 degrees is -√2.