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This will be the case with all the restricted ranges that follow.Īnother notation for arcsin x is sin −1 x.
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To restrict the range of arcsin x is equivalent to restricting the domain of sin x to those same values. Angles whose sines are negative will fall in the 4th quadrant. Īngles whose sines are positive will be 1st quadrant angles. The range, then, of the function y = arcsin x will be angles that fall in the 1st and 4th quadrants, between − and. The angle whose sine is − x is simply the negative of the angle whose sine is x. The angle of smallest absolute value falls in the 4th quadrant between 0 and −. Angles whose sines are negative fall in the 3rd and 4th quadrants. The first quadrant angle is the angle with the smallest absolute value whose sine is ½. They are called the principal values of y = arcsin x. How will we do that? We will restrict them to those angles that have the smallest absolute value. Therefore we must restrict the range of y = arcsin x - the values of that angle - so that it will in fact be a function so that it will be single-valued. It wll be any angle whose corresponding acute angle is. Now there are many angles whose sine is ½. Because in the unit circle, the length of that arc is the radian measure. Strictly, arcsin x is the arc whose sine is x. ½, then the challenge is to name the radian angle x.Īrcsin x is the angle whose sine is the number x. Inversely, if we are given that the value of the sine function is Thus if we are given a radian angle, for example, then we can evaluate a function of it. T HE ANGLES in calculus will be in radian measure.
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Inverse trigonometric functions - Topics in trigonometry