Correct. Incorrect: Since neither arctan nor arcsin are functions, there are infinitely many values for both these angles. One counterexample is: sin(60 + 30) = sin(90) = 1 Use what you know about trigonometric functions and their inverses to solve these simple equations. Two examples are included for you. (Unit circles can also be useful.) Example 1: Example 2: ArcCos sin (cos-1 1 + tan-1 1) = x The answers will be an angle. Use The answer will be a number, not an to remind yourself angle. Simplify parentheses first. θ Let Arccos = = cos-1 1 = tan-1 1 θ θ1 θ2 = 0 = 45 Ask yourself, what angle has a cos θ1 θ2 value of . sin (0 + 45) = x Cos Substitution θ Using the definition of Arccos. sin(45) = x = = x θ Why included? isn’t So, ArcCos = So, sin (cos-1 1 + tan-1 1) = = = Arcsin 1 θ θ Sin =1 θ Cos = θ Using the definition of Arcsin. Using the definition of Arccos. = θ = θ sin-1 2 = x cos ( tan-1 sin-1 ½ ) = x –
Correct. Incorrect: Since neither arctan nor arcsin are functions, there are infinitely many values for both these angles. One counterexample is: sin(60 + 30) = sin(90) = 1 Use what you know about trigonometric functions and their inverses to solve these simple equations. Two examples are included for you. (Unit circles can also be useful.) Example 1: ArcCos The answers will be an angle. Use θ to remind yourself Let Arccos = θ Ask yourself, what angle has a cos value of . Cos θ Using the definition of Arccos. = θ Why isn’t included? So, ArcCos = Example 2: sin (cos-1 1 + tan-1 1) = x The answer will be a number, not an angle. Simplify parentheses first. θ1 = cos-1 1 θ2 = tan-1 1 θ1 = 0 θ2 = 45 sin (0 + 45) = x Substitution sin(45) = x = x So, sin (cos-1 1 + tan-1 1) = θ = Cos θ = Using the definition of Arccos. θ = θ = Arcsin 1 Sin θ =1 Using the definition of Arcsin. θ = sin-1 2 = x cos ( tan-1 – sin-1 ½ ) = x