find all values of theta in the interval (0,360) that have the given value:
sintheta= √2
______
2
Good explanation, but I think you want 45 deg.
To find
sin(θ)=(√2)/2
between 0 and 360°
where
(√2)/2 = 0.707 (approx.),
it would be much more visual to look at the form of the sine function between 0 and 360°.
Study the graph at the link (at the end of this post).
It is positive between 0 and 180° and negative for the rest of the interval, therefore between 180 and 360° there will not be a solution.
Between 0 and 180°, draw an imaginary horizontal line through y=0.7. This line will intersect the graph at two places, and precisely, at 60° and 120°.
The axis of symmetry is at 90°, so it is easy to remember (90-30)°=60°, and (90+30)°=120°.
You will use these angles very often in trigometry for the rest of your academic career, so it's worthwhile memorizing them.
http://www.google.ca/imgres?q=graph+sine+function&hl=en&sa=X&biw=1071&bih=625&tbm=isch&prmd=ivns&tbnid=OMu42wKsarjpBM:&imgrefurl=http://www.wsd1.org/waec/math/pre-calculus%2520advanced/Trigonometry/Graphing/graphingintro.htm&docid=ZDN6zUNT4oa-KM&w=641&h=341&ei=GgtmTrTcBaLs0gHoy6HRDQ&zoom=1&iact=hc&vpx=722&vpy=233&dur=1&hovh=164&hovw=308&tx=255&ty=121&page=2&tbnh=103&tbnw=194&start=12&ndsp=12&ved=1t:429,r:7,s:12
To find all values of theta in the interval (0, 360) that satisfy the equation sin(theta) = (√2)/2, we can use the unit circle and apply the properties of trigonometric functions.
First, let's recall the special angles on the unit circle where sin(theta) = (√2)/2. These angles are 45 degrees and 225 degrees.
Since the question asks for values in the interval (0, 360), we need to consider the positive angles that are less than 360 degrees but greater than 0 degrees. Namely, we need to consider 45 degrees, 135 degrees, 225 degrees, and 315 degrees.
So, the values of theta in the interval (0, 360) that satisfy sin(theta) = (√2)/2 are 45 degrees, 135 degrees, 225 degrees, and 315 degrees.
To find all values of theta in the interval (0, 360) that satisfy the equation sin(theta) = √2/2, we can use the knowledge of the unit circle and the trigonometric functions.
Step 1: Identify the reference angle
The reference angle is the angle between the terminal side and the x-axis in the first or fourth quadrant. In this case, sin(theta) = √2/2 is positive, so the reference angle will be in the first quadrant. Looking at the unit circle, we can see that the reference angle for sin(theta) = √2/2 is 45 degrees or π/4 radians.
Step 2: Determine the values in the given interval
Since sin(theta) is a periodic function, it repeats its values every 360 degrees or 2π radians. We need to find all the angles in the interval (0, 360) that have the reference angle of 45 degrees or π/4 radians.
In the first quadrant, the angle with a reference angle of 45 degrees is 45 degrees itself.
To find the angles in the second, third, and fourth quadrants, we need to consider the patterns of sin(theta) on the unit circle.
In the second quadrant, the sine function is positive for angles between 0 and 180 degrees. So in this case, the angle would be 180 degrees - 45 degrees = 135 degrees.
In the third quadrant, the sine function is negative, so the angle would be 180 degrees + 45 degrees = 225 degrees.
In the fourth quadrant, the sine function is negative. The angle would be 360 degrees - 45 degrees = 315 degrees.
So the values of theta in the interval (0, 360) that satisfy sin(theta) = √2/2 are 45 degrees, 135 degrees, 225 degrees, and 315 degrees.