Math 166 CO1 Lesson 1.9 Inverse Trigo & Trigo Equations PDF

MATH 166 College Trigonometry with Solid Mensuration Trigonometry 11e by R. Larson  Course Outcome 1 Fundamentals of Trigonometry CO1 – Lesson 1.9 Inverse Trigonometric Functions and Trigonometric Equations  Lesson Objectives At the end of this lesson, students should b...

MATH 166 College Trigonometry with Solid Mensuration Trigonometry 11e by R. Larson  Course Outcome 1 Fundamentals of Trigonometry CO1 – Lesson 1.9 Inverse Trigonometric Functions and Trigonometric Equations  Lesson Objectives At the end of this lesson, students should be able to: 1. define and understand the concept of inverse trigonometric functions, including their restricted domains and principal value ranges. 2. evaluate inverse trigonometric functions at specific values and express them in terms of their corresponding angles. 3. solve trigonometric equations involving both basic trigonometric functions and inverse trigonometric functions. 4. apply inverse trigonometric functions to solve real-world problems, such as finding angles of elevation or depression. 5. compose trigonometric and inverse trigonometric  Introduction In Lesson 1.2, we defined the trigonometric functions as ratios of the sides of a right triangle. For example, given the triangle on the right, we have ; In this lesson, we are going to define an operation or a function that will find the value of the angle that can yield the indicated trigonometric value or ratio. Now, as determined in Lesson 1.3, the special angle satisfies and So, we can say that the angle in the triangle is We then define the concept of inverse trigonometric functions to serve as the universal language for finding angles or real numbers whose trigonometric values are given.  Inverse Trigonometric Functions Remarks: 1. The notations and mean exactly the same. 2.. 3. The restriction in the range ensures that the operation defines a function (i.e., no multiple values of for each value of.  Examples Find the exact value of the following: a) Let , where with. Since (special angle), it follows that b) Let , where with. Since , it follows that  Sine-Inverse Sine Identity Find the exact value of the following: a) Since ,. b) is not in , so we cannot apply outright the definition, that is,.  Inverse Cosine Function Find the exact value of the following: a) Let , where and. Based from values of special angles,. Thus, b) Since and. Then.  Cosine-Inverse Cosine Identity Find the exact value of the following: a) Since ,. b) is not in , so..  Inverse Tangent Function Find the exact value of the following: a) Since and it follows that b) Use the fact that where. Then.  Tangent-Inverse Tangent Identities Find the exact value of the following: a) Answer: b) is not in. The first identity requires that the value of is in QI or QIV. Since tangent is negative in QII (where lies), and then we need to find in QIV in which. Using special angles,. But is also not in. To resolve this, we trace in clockwise direction (negative), which is (co-terminal). Then.  Examples Evaluate. Solution: Let. Refer to the figure below. Then 𝟑 𝟐 𝜽 √ 32 −22 =√ 5  Examples Evaluate: Solution: √ 52 +122=13 12 Let and. 𝜽 From the figures on the right, , 5 and. 5 3 𝜷 √ 52 −32 =4 Exercise 1. Find the exact value of the following: a) b) c) d) e) f) g) h) i) 2. Evaluate the following: a) b) c)  Trigonometric Equations Previously, we considered trigonometric equations called identities. These equations are true for all replacements of the variable(s) for which both sides are defined. We now consider another class of equations, called conditional equations, which may be true for some replacements of the variable(s) but false for others. For example, is a conditional equation, since it is true for and false for.  Trigonometric Equations To solve a trigonometric equation, it is best to write the equation using a single trigonometric function. This can be done by a) using appropriate identities b) using algebraic manipulations such as factoring, combining fractions, etc. The equation can be solved by applying the “trigo-inverse trigo” identities. In some cases the interval from which the solution values may be taken is specified. Otherwise, obtain the principal value based from the domain of the inverse function then add (for sine or cosine) or (for tangent), , to get the general solution.  Example Solve the equation. Solutions: For convenience, we can Method 2: Method 1: transform first the equation into algebraic form by letting. Then Since it is specified that , we can consider in QIII and QIV, particularly those that are multiples of. Then This yields the same solution: In this example, the interval for the value/s of was already specified. Thus limiting the values into just 2 values.  Example Find all solutions of the equation. Solution: This is almost the same as the previous example, only that we are now asked to give all possible solutions. Using the same process, we have In QI, is satisfied by the special angle (equivalently, ). Since , or. In QIII, the angle whose reference angle is is [recall: in QIII, and so ]. In QIV, the angle with reference angle is [use: in QIV]. Therefore the solution is  Example Use inverse trigonometric functions to solve the equation. Solution: As in previous solutions, simplification of the given equation reduces to Applying the inverse sine function: Note that is coterminal with. Since the range of the inverse function is we have to consider as the unique solution.  Example Solve the equation. Solution: This equation is quadratic in. Let. Noting that the product becomes 0 only if either factor is 0, we must have Thus the complete solution is:  Exercise 1. Solve the equation for. 2. Solve the equation for. 3. Find all solutions of the following equations (in radians): a) b) 4. Use inverse trigonometric functions to solve the following equations: a) b) c) 5. If a 6-foot lamppost makes an 8-foot shadow on the sidewalk, find its angle of elevation to the sun (answer in nearest degree). Resources