In Class Problems Section 5.6

#5 If possible find the exact value.

1. tan[arctan(-5)] = -5. Since -5 is in the domain of arctan the identity property of the composition of inverse functions applies.
2. arcsin(sin 5p /3) = arcsin(sin -p /3) = -p /3. Here we first find a coterminal angle in the range of the arcsine function. This angle is -p /3. Now the identity property of the composition of inverse functions applies.
3. cos(cos^-1 p ) = undefined. p is not in the domain of the arccosine function.

#10 Evaluate the expression without the aid of a calculator.

1. arccos(-sqrt(3)/2) = 2p /3. Here we see that -sqrt(3)/2 is in the domain of the arccosine function, in the second quadrant. adj/hyp Þ q is 120^o or 2p /3.
2. arcsin(-sqrt(2)/2) = -p /4. Here we see that -sqrt(2)/2 is in the domain of the arcsine function, in the fourth quadrant. opp/hyp Þ q is -45^o or -p /4

#24 Use an inverse trigonometric function to write q as a function of x.

The inverse function we are looking for must relate the side adjacent and the hypotenuse. The cosine of q is the ratio of the side adjacent to the hypotenuse. q (x) = arccos(4/x), or q (x) = arcsec(x/4).

#28 tan(arctan25) = 25. Since 25 is in the domain of the arctangent function, we apply the identity property of inverse functions.

#55 arctan(9/x) = arcsin(9/sqrt(9² + x)), x ¹ 0.

After determining the length of the hypotenuse, we see that opp/hyp is 9/sqrt(9^2 + x).

Essential Trigonometry, Math Links

by Jack Tompkins