Inverse trig functions create right triangles. Elementary Functions. Some Worked Problems on Inverse Trig Functions

Transcription

1 Inverse trig functions create right triangles An inverse trig function has an angle (y or ) as its output That angle satisfies a certain trig expression and so we can draw a right triangle that represents that expression One can always draw a right triangle with an inverse trig function and think of the output as a certain angle in that triangle Part, Trigonometry Lecture 7a, Solving Problems with Inverse Trig Functions For example, the equation arcsin() = implies that sin = and so corresponds to a right triangle with hypotenuse, with one of the acute angles and the length of the side opposite Dr Ken W Smith Sam Houston State University 0 0 / 7 0 / 7 Some Worked Problems on Inverse Trig Functions Some Worked Problems on Inverse Trig Functions We will practice this idea with some worked problems When we work with inverse trig functions it is especially important to draw a triangle since the output of the inverse trig function is an angle of a right triangle Draw a right triangle with the appropriate lengths and use that triangle to find the sine of the angle if cos() = cos() = cos() = 08 cos() = 06 Partial solutions If cos() = then draw a triangle with legs of length, and If cos() = then the sine of is then draw a triangle with legs of length, and In a similar manner, we can simplify tan(arcsin ) to tan(arcsin()) = If cos() = 08 then draw a triangle with legs of length, and hypotenuse of length The sine of is hypotenuse of length The sine of is hypotenuse of length If the cosine of is Indeed, one could think of inverse trig functions as creating right triangles The angle in the drawing below is arcsin() Notice that the Pythagorean theorem then gives us the third side of the triangle (written in blue); its length is This allows us to simplify expressions like cos(arcsin ), recogniing that p cos(arcsin ) = cos() = If cos() = 06 then draw a triangle with legs of length, and hypotenuse of length The sine of is 0 / 7 0 / 7

2 Some Worked Problems on Inverse Trig Functions Some worked problems Simplify (without use of a calculator) the following expressions arcsin[sin( π 8 )] arccos[sin( π 8 )] cos[arcsin( )] Solutions Since arcsin is the inverse function of sine then arcsin[sin( π 8 )] = π 8 If is the angle π 8 then the sine of is the cosine of the complementary angle π π 8, which, after getting a common denominator, simplifies to π 8 In other words, the sine of π 8 is the cosine of π 8 so arccos[sin( π 8 )] = π 8 (Notice that I ve solved this problem this without ever having to figure out the value of sin( π 8 )) To simplify cos[arcsin( )] we draw a triangle with hypotenuse of length and one side of length, placing the angle so that sin() = The other short side of the triangle must have length 8 = by the Pythagorean theorem so the cosine of is So cos[arcsin( )] = 0 / 7 Some worked problems Simplify (without the use of a calculator) the following expressions: arccos(sin()), assuming that is in the interval [0, π ] Solutions To simplify arccos(sin()), we draw a triangle (on the unit circle, say) with an acute angle and short sides of lengths x, y and hypotenuse x π The sine of is then y and the arccosine of y must be the complementary angle π So arccos(sin()) = π 0 6 / 7 Some worked problems y Simplify arccos(y) + arcsin(y) Solution Notice in the triangle in the figure below, that the sine of is y and the cosine of π is y π y In the next presentation, we will solve more problems with inverse trig functions x (End) So arcsin(y) = and arccos(y) = π Therefore arccos(y) + arcsin(y) = + ( π ) = π Indeed, the expression arccos(y) + arcsin(y) merely asks for the sum of two complementary angles! By definition, the sum of two complementary angles is π! 0 7 / / 7

3 Some problems involving inverse trig functions include the composition of the inverse trig function with a trig function If the inverse trig function occurs first in the composition, we can simplify the expression by drawing a triangle Worked problems Do the following problems without a calculator Find the exact value of sin(arccos( )) tan(arcsin( )) Part, Trigonometry Lecture 7b, Inverse Trig Expressions Create Triangles Dr Ken W Smith Solutions To compute sin(cos ( )) draw a triangle with legs, 7 and hypotenuse The angle needs to be in the second quadrant so the cosine will be negative In this case, the sine will be positive So the Sam Houston State University 0 sine of the angle should be 0 9 / 7 sin( tan ( )) Solution To compute = sin cos where tan() = draw a triangle with legs, and hypotenuse The cosine of the angle is and the sine of the angle is Since the original problem has a negative sign in it, and we are working with the arctangent function, then we must be working with an angle in the fourth quadrant, so the sine is really Now we just plug these values into the magical identity given us: sin() = sin cos = ( )( ) = 0 To compute tan(sin ( )) draw a triangle with legs, 7 and hypotenuse The tangent of the angle should be 7 But the 0 angle is in the fourth quadrant so the final answer is 7 0 / 7 Simplify sin( arctan( )) (Use the trig identity sin = sin cos ) 7 / 7 Simplify the following expressions involving arctangent: tan(arctan()), sin(arctan()), cot(arctan()), sec(arctan()) Solutions To compute tan(arctan()) just recognie that tan and arctan are inverse functions and so tan(arctan()) = To compute sin(arctan()) draw a right triangle with sides, and hypotenuse + The sine of the angle is + In the figure above, the cotangent of the angle is The secant of the angle should be + 0 / 7

4 Just like other functions, we can algebraically manipulate expressions to create an inverse function Some worked problems Find the inverse function of y = sin( x) + Solutions To find the inverse function of y = sin( x) +, let s exchange inputs and outputs: x = sin( y) + and then solve for y by subtracting from both sides applying the arcsin to both sides, and then squaring both sides x = sin( y), arcsin(x ) = y (arcsin(x )) = y Find the inverse function of y = sin( x + ) Solutions We set x = sin( y + ), take the arcsine of both sides: arcsin(x) = y + ), square both sides (arcsin(x)) = y +, and then subtract from both sides The inverse function of y = sin( x + ) is y = (arcsin x) so that the answer is is y = (arcsin(x )) 0 / 7 0 / 7 Find the inverse function of y = e sin( x+) Solutions We set x = e sin( y+) take the natural log of both sides: ln(x) = sin( y + ), then take the arcsine of both sides arcsin(ln(x)) = y +, and then subtract from both sides arcsin(ln(x)) = y, and finally square both sides Find the inverse function of y = sin(arccos x) Solutions First we simplify sin(arccos x) Draw a right triangle with a hypotenuse of length and an acute angle with adjacent side of length x The side opposite of has length (by the Pythagorean theorem) x So the cosine of is just x We have simplified y = sin(arccos x) to y = x It happens that the inverse function of y = x obeys the equation x = y so x = y so y = x so y = x (That is y = x is its own inverse function!) The inverse function of y = e sin( x+) is y = (arcsin(ln x) ) 0 / / 7

5 Using inverse trig functions REMEMBER: When faced with an inverse trig function, think about the triangle the function creates! In the next presentation, we will look at trig identities and equations (End) 0 7 / 7