Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
MA 15400 Lesson 16 Section 7.6 The Inverse Trigonometric Functions Find θ in the interval [0, 2π) for each statement. 1 3 tan sin 3 2 7 2 , , 6 6 3 3 1 cos 2 7 4 , 4 While this is true for equations with the directions ‘Find all solutions of the equation in the interval [0, 2)’, today we are going to define the inverse trigonometric functions. INVERSE TRIGONOMETRIC FUNCTIONS: Remember that a function must pass a vertical line test. And, for each x, there is only one y. For a trigonometric function, each angle yields only one trig value (a ratio). For an inverse trigonometric function, each trig value (ratio) would have to yield only one angle. However, we 1 can see from above, a trig value, such as can correspond to more than one angle, for 3 7 and example both . Your calculator is ‘set up’ to return only one angle for each trig 6 6 value. Remember that the inverse sine key or inverse tangent key only return a value in quadrants I or IV. The inverse cosine key only returns an angle in quadrants I or II. Notation for Inverse Trigonometric Functions: y = arcsin(x) = sin-1(x) = θ y =arccos(x) = cos-1 (x)= θ y = arctan(x) = tan-1(x) = θ 1 1 arctan tan 1 3 3 6 3 3 sin 1 arcsin 2 3 2 1 1 arccos cos 1 2 2 4 Why only one answer? Remember you calculators only returned values in certain quadrants when we were working with inverce functions? arcsin is only defined from to inclusively, which are 2 2 quadrants I and IV. arctan is also only defined from to , which are quadrants I and IV. 2 2 However, arccos is defined from 0 to inclusively, which are quadrants I and II. This is reinforced when you use the inv tan, inv cos, or inv sin keys on you calculator. sin-1 tan-1 sin-1 tan-1 cos-1 cos-1 Notice: For each function, one quadrant has positive values and the other quadrant has negative values. 1 Remember: An inverse trig function returns an angle. Given a trig value as x (a ratio), it returns a y that is an angle! MA 15400 Lesson 16 Section 7.6 The Inverse Trigonometric Functions Summary: arcsin is defined in a range of , and domain of [1,1] 2 2 arctan is defined in a range of , (Remember, at and , the tangent is undefined) and 2 2 2 2 domain of all real numbers arccos is defined in a range of 0, and domain of [1,1] Find the exact value of the expression whenever it is defined. 1 sin 1 2 2 cos 1 2 tan 1 1 1 arcsin 2 3 arccos 2 arctan 3 For the following problems: Always work from the inside out!!! 2 sin arcsin tan tan 1 5 cos[cos 1 (1)] 2 You will notice when the trig function is on the ‘outside’, the two functions ‘eliminate’ each other and the result is the trig value. This may not happen when the inverse function is on the ‘outside’. Therefore, always work from the inside out. 2 MA 15400 Lesson 16 Section 7.6 The Inverse Trigonometric Functions 5 arcsin sin 4 5 arccos cos 4 5 arctan tan 4 Notice: In the above 3 examples, the argument in each was 5 . However, none of the answers were 5 . It is outside of 4 4 the range. Also, notice that the 3 answers are not the same. 3 sin cos 1 2 1 cos tan 1 3 1 tan sin 1 2 Find the exact value of the expression whenever it is defined. 5 sin 2 arccos 13 sin 2 2sin cos 5 arccos 13 T 5 cos 13 The angle θ is in Q II. 4 cos 2 sin 1 5 3 4 4 sin 5 The angle θ is in QI sin 1 5 MA 15400 Lesson 16 Section 7.6 The Inverse Trigonometric Functions 9 tan 2 tan 1 40 Write the expression as an algebraic expression in x for x > 0. x sin 1 cos sin x 1 sin 2 arccos x cos sin 2 1 x2 1 1 cos x 1 x θ b 1 θ x x 2 b 2 12 b2 1 x2 1 x2 b 1 x2 Proof (using a calculator) cos(sin 1 x) 1 x 2 Let x = 0.26 cos(sin 1 (0.26) 1 (0.26) 2 cos(0.263022203) 1 0.0676 0.9656086 0.9324 0.9656086 0.9656086 4 MA 15400 Lesson 16 Section 7.6 The Inverse Trigonometric Functions Graph: Most function graphs continue infinitely. This graph ‘fits’ in a rectangle. 2 y sin 1 x Trig Value Angle -1 2 3 3 2 1 4 2 1 2 6 -1 0 0 1 2 6 1 4 2 3 3 2 1 2 y tan 1 x 3 4 6 1 4 6 3 2 2 3 4 6 2 -2 6 4 3 2 5 MA 15400 Lesson 16 Section 7.6 The Inverse Trigonometric Functions y cos1 x π 5 6 3 4 2 3 2 3 4 6 -1 1 1 y cos 1 x 2 π 5 6 3 4 2 3 2 3 4 6 -1 6 1 MA 15400 Lesson 16 Section 7.6 The Inverse Trigonometric Functions y cos1 (2 x) π 5 6 3 4 2 3 2 3 4 6 -1 1 7