By trial and error with your calculator, find the angle theta for which sin theta and theta differ by 5%. This calculation must be done in radians. When you have found theta, express it in degrees.
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try 15 degrees, compute sin15/tan15. If the answer is between .95 and 1.05, increase the angle by 2 degrees. You will quickly find the angle.
To find the angle theta for which sin theta and theta differ by 5%, you can use trial and error with a calculator. Here's the step-by-step process to achieve this:
1. Start by choosing an angle in radians. Let's assume an initial guess of theta = 1 radian.
2. Calculate the value of sin theta using your calculator.
3. Calculate the difference between sin theta and theta: difference = |sin theta - theta|.
4. If the difference is equal to 0.05 or within a small range around it (e.g., between 0.045 and 0.055), then you have found the angle. If not, continue to step 5.
5. Adjust your angle estimate by a small increment (e.g., 0.001 radian) depending on whether the difference is greater or smaller than 0.05.
6. Repeat steps 2-5 until you find an angle theta that satisfies the condition.
Let's go through the process step by step:
1. Assume an initial guess: theta = 1 radian.
2. Calculate the value of sin(1) using your calculator: sin(1) ≈ 0.84147.
3. Calculate the difference: difference = |sin(1) - 1| = 0.15853.
4. Since the difference is not within the desired range (0.045-0.055), we move to step 5.
5. Adjust the angle by a small increment, let's say 0.001 radian: theta = theta + 0.001 = 1 + 0.001 = 1.001 radians.
6. Repeat steps 2-5 with the new angle estimate.
Continue this trial and error process until you find an angle theta that satisfies the condition, where the difference is within the desired range. Once you have found theta in radians, you can convert it to degrees by multiplying by 180/π (approximately 57.2958).
I hope this helps! Let me know if you have any further questions.
To find the angle theta for which sin theta and theta differ by 5%, we can use a trial and error approach with a calculator. Here's how you can do it:
1. Start by selecting an initial guess for theta, let's say theta = 1 radian (you can choose a different value if you prefer).
2. Calculate the value of sin theta using your calculator.
3. Calculate the difference between sin theta and theta.
4. Check if the difference is equal to 5% of theta. If not, adjust the value of theta and repeat steps 2 and 3 until you find theta.
To express theta in degrees, after finding the value of theta in radians, you can use the conversion formula:
Degrees = Radians * (180 / π)
Now, let's go through an example:
1. Let's start with theta = 1 radian.
2. Calculating sin(1) on a calculator gives approximately 0.841.
3. The difference between sin(1) and 1 is 0.841 - 1 = -0.159.
4. Since -0.159 is not equal to 5% of 1 (0.05 * 1 = 0.05), we need to adjust the value of theta.
- If the difference is negative, increase the value of theta.
- If the difference is positive, decrease the value of theta.
Let's try increasing theta to 1.1 radians:
1. Set theta = 1.1 radian.
2. Calculating sin(1.1) gives approximately 0.891.
3. The difference between sin(1.1) and 1.1 is 0.891 - 1.1 = -0.209.
4. Since -0.209 is not equal to 5% of 1.1 (0.05 * 1.1 = 0.055), we need to adjust theta again.
Repeat steps 1-4 with different values of theta until you find an angle for which the difference is within 5% (or very close to it).