Rianna H. answered • 12/30/20
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Hello Oliver,
a) sin(θ)= -1/2
θ = 30° or π/6
However, since the value is negative, our angle cannot fall within quadrants 1 or 2. So our answer are 7π/6 and 11π/6
b) sin(θ) = 1/√2 or √2/2
θ = 45° or π/4
Since our value is positive, our angle falls within quadrants 1 and 2. So our answers are π/4 and 3π/4
I hope this helps!
Rianna H.
Sure! I got these values all from the unit circle. It's best to memorize the unit circle and the sin, cos and tan values of 30 deg, 45 deg, and 60 deg. You can just google unit circle and look under images. From the circle I know that sin(30deg) is 1/2 and 30 deg in radian form is pi/6. Next, because they want the values of sin(theta) that equal -1/2, we have to find the values of pi/6 that fall within the quadrants that give us negative sin values. These quadrants are 3 and 4; I use the All Students Take Calculus acronym to remember the specific quadrants. So, from the interval [0,2pi] 7pi/6 (in quad 3) and 11pi/6 (in quad 4) are my answers.
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12/30/20
Oliver S.
thank you so much, and sorry just one more thing, still confused on finding the angles, can you try to explain how to do it one more time in more detail. I've never used the All students Take Calculus acronym before.
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12/30/20
Rianna H.
No worries! The unit circle shows us what the sin and cos of 30deg, 45deg and 60deg are between [0,2pi]. In the very first quadrant, we can see pi/6 (30deg), pi/4 (45deg) and pi/3 (60deg) and their corresponding sin and cos values which are written as (x,y) values. The x value is always what the cos(theta) is and the y values is the sin(theta) value. I remember this by the word "cousins" or (Cos, sins). In quadrants 2, 3 and 4, we are dealing with the related angle values of 30, 45 and 60 degrees or the angles created from the origin line bordering that particular quadrant. Each quadrant continues to have values for 30, 45 and 60 deg but because the location changes, then the radian values also change. For 30 deg: Q1-pi/6 Q2- 5pi/6 Q3- 7pi/6 Q4- 11pi/6 For 45 deg: Q1- pi/4 Q2- 3pi/4 Q3- 5pi/4 Q4- 7pi/4 For 60 deg: Q1- pi/3 Q2- 2pi/3 Q3- 4pi/3 Q4- 5pi/3 Now, with the "All Students Take Calculus" acronym, I can figure out which quadrant I am in. Q1- "All" here all values of sin and cos are positive Q2- "Students" here only SIN values are positive Q3- "Take" here only TAN values are positive Q4- "Calculus" here only COS values are positive So back to your original question, sin(theta)= -1/2. From the unit circle I know that sin(30) is always 1/2, this is how I know the angle theta is 30deg or pi/6. Now, because of the negative sign, the radian value must be one that falls in a quadrant that provides negative sin values. These quadrant are 3 and 4. In these quadrants, the related angles that measure 30deg and 7pi/6 and 11pi/6. Does it make a little more sense now? If it doesn't, just let me know! :)
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12/31/20
Oliver S.
that makes so much sense, thank you, the unit circle is a lot easier to use, we weren't taught to use that :)
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12/31/20
Oliver S.
Can you explain how you got 30 degrees and pi/6? and 7pi/6 and 11pi/6, thanks.12/30/20