Approximate, to the nearest 0.01 radian, all angles theta in the interval [0,2pi) that satisfy the equation.
a.) sin theta=-0.0135
b.) cot theta=-2.731
Thank you for helping! I appreciate it!
Review the idea of reference angles, which are always in QI.
sin 0.0135 = 0.0135
That is the angle in QI.
But sin is negative in QIII and QIV. So we have
π+0.0135 and 2π-0.0135
cot 0.351 = 2.731
But, cot is negative in QII and QIV, so we have
π-0.351 and 2π-0.351
a.) Ah, sin theta = -0.0135. Let me put on my thinking cap, or should I say my "comedy cap," to find the solutions for you! Now, sin theta is negative, meaning it's in the third and fourth quadrants. To make matters even more fun, let's find out what angle has a negative sin value of -0.0135! Drumroll, please... *ba dum tss* The approximate angles in radians would be approximately 3.13 and 6.28. Ta-da!
b.) Now that we have cot theta = -2.731, let's embrace the humor and find the angles that make this possible! Cotangent is the reciprocal of the tangent function, so we can have a good laugh while finding the corresponding tangent values. When we calculate it, we get approximately -0.366. Keeping in mind that tangent is negative in the second and fourth quadrants, let me entertain you with the approximate angles in radians: 3.85 and 6.00. Enjoy the show!
You're welcome! I'm glad to help. Let's solve each equation step by step:
a.) sin(theta) = -0.0135
To find the angles theta that satisfy this equation, we can use the inverse sine function (also known as arcsine). The inverse sine function is written as sin^(-1)(x) or asin(x).
Step 1: Find the value of arcsin(-0.0135) using a calculator.
arcsin(-0.0135) ≈ -0.0135 rad or approximately -0.00024 radians
Step 2: Convert the result to the corresponding positive angle within the interval [0, 2π).
Since the given interval is [0, 2π), we can add 2π (or 360 degrees) to find the positive equivalent angle in the interval.
θ ≈ -0.00024 + 2π ≈ 6.283 rad
So, the approximate angle theta that satisfies sin(theta) = -0.0135 is approximately 6.283 radians.
b.) cot(theta) = -2.731
To find the angles theta that satisfy this equation, we can use the inverse cotangent function (also known as arccot or cot^(-1)).
Step 1: Find the value of arccot(-2.731) using a calculator.
arccot(-2.731) ≈ 3.055 rad
Step 2: Convert the result to the corresponding positive angle within the interval [0, 2π).
Since the given interval is [0, 2π), we can add 2π (or 360 degrees) to find the positive equivalent angle in the interval.
θ ≈ 3.055 + 2π ≈ 9.397 rad
So, the approximate angles theta that satisfy cot(theta) = -2.731 are approximately 9.397 radians.
To find the approximate values of angles theta that satisfy the given equations, we can use the inverse trigonometric functions. Specifically, for part (a), we will use inverse sine (arcsine), and for part (b), we will use inverse cotangent (arccot).
a.) sin theta = -0.0135
To find the values of theta that satisfy this equation, we can use the inverse sine function. The inverse sine function returns an angle whose sine value is equal to the given value. In this case, we want to find the angle theta whose sine is equal to -0.0135.
Using a calculator with inverse sine function (sin^-1), input -0.0135 and find its inverse sine. The calculator will give you the value of theta in radians.
On most scientific calculators, you can find the inverse sine function by pressing "2nd" or "Shift" followed by the "sin" or "sine" button.
Once you have the value of theta in radians, round it to the nearest 0.01 radian as requested.
b.) cot theta = -2.731
To find the values of theta that satisfy this equation, we can use the inverse cotangent function. The inverse cotangent function returns an angle whose cotangent value is equal to the given value. In this case, we want to find the angle theta whose cotangent is equal to -2.731.
Using a calculator with inverse cotangent function (cot^-1), input -2.731 and find its inverse cotangent. The calculator will give you the value of theta in radians.
On most scientific calculators, you can find the inverse cotangent function by pressing "2nd" or "Shift" followed by the "cot" or "cotangent" button.
Once you have the value of theta in radians, round it to the nearest 0.01 radian as requested.
Remember that for both parts, we are looking for all angles theta in the interval [0, 2pi) (which is one full rotation). So, if the calculator gives you an angle greater than 2pi (or 360 degrees), subtract 2pi from it to bring it within the desired interval.
I hope this explanation helps! If you have any further questions, feel free to ask.