Pre-Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on a Trigonometric Expression The FUNdamental TRIGonometric Identities In trigonometry, there are expressions and equations that are true for any given angle. These are called identities. An infinite number of trigonometric identities exist, and we are going to prove many of these identities, but we are going to need some basic identities first. The six basic trig ratios will lead to our first identities sin θ = csc θ = cos θ = sec θ = tan θ = cot θ = And our knowledge of Pythagoras will determine the remaining FUNdamental TRIGonometric Identities Pythagorean Identities: sin cos 1 1 tan sec 1 cot csc Reciprocal and Quotient Identities: sec 1 cos csc 1 sin cot 1 tan tan sin cos cot cos sin
Pre-Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Corollary Identities ( a statement that follows readily from a previous statement) sin cos 1 1 tan sec 1 cot csc Simplifying a Trigonometric Expression There are many different strategies to simplifying a trigonometric expression. The following examples will look at the most common types of strategies. Write as a fraction with a common denominator Factor as a difference of squares Change everything to sine and cosine. Multiply by the conjugate Now we can use these strategies along with the eight fundamental identities to simplify expressions
Pre-Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Example 1: Simplify Example 2: Simplify Example 3: Simplify Example 4: Simplify Example 5: Simplify Example 6: Simplify
Pre-Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Restrictions Just like any algebraic expression, a trigonometric expression cannot have zero in the denominator. We must consider the exact values that would result in a denominator of zero. Example 7: Determine the restrictions on for Practice: Page 213 #2-9 (as many as needed)
Pre-Calculus Mathematics 12 6.2 Verifying Trigonometric Identities Goal: 1. Verify and prove Trigonometric Identities Trigonometric Identities When verifying trigonometric identities, the key is using the rules for algebra as well as the fundamental trigonometric identities to rewrite and simplify expressions. An identity has been proven when the right side of the equal sign is the same as the left side of the equal side. Example 1: Prove the identity: Example 2: Prove the identity:
Pre-Calculus Mathematics 12 6.2 Verifying Trigonometric Identities Example 3: Prove the identity: Example 4: Prove the identity: Practice: Page 271 #1-26
Pre-Calculus Mathematics 12 6.3 Trigonometric Equations Goal: 1. Solve trigonometric equations for conditional statements and general form 2. Solve equation with angles other then θ or x. Trigonometric Equations A trigonometric equation is different from a trigonometric identity in that the equation is true for some values of the variable and not all values. When solving a trig equation, we can solve for a specified domain of values ( usually ) or in general form ( for all potential values) Example 1: Solve: for a) b) general form Example 2: Solve: for a) b) general form Example 3: Solve: for a) b) general form
Solving trigonometric equations with angles other than θ Pre-Calculus Mathematics 12 6.3 Trigonometric Equations Solving a trig equation for angles with coefficients ( 2θ, 3θ, etc) follows very similar steps: 1. Isolate the trig function. eg. sin 2θ 2. Determine the general solution(s) for angle 2θ 3. Determine the general solution(s) for angle θ. (divide by the coefficient) 4. Determine the specific ( conditional ) solution(s) for the given interval Example 4: Solve: a) general form b) for Example 5: Solve: a) general form b) for Example 6: Solve: for a) b) general form Practice: Page 281 #1-5
Pre-Calculus Mathematics 12 6.4 Sum and Difference Identities Goal: 1. Use sum and difference identities to solve complex trig problems 2. Simplify expressions and prove identities involving sums and differences Sum and Difference Identities Identities are not limited to the fundamental identities and single angles. We can also use identities involving sums and differences. The derivations of these identities are shown on page 287 of your textbook. We will be looking not at proving these identities but using these identities. cos cos cos sin sin cos cos cos sin sin tan tan sin sin cos cos sin sin sin cos cos sin tan tan tan tan 1 tan tan 1 tan tan Example 1: Find the exact value: Example 2: Find the exact value: Example 3: Express as a single function, then evaluate. Example 4: Express as a single function, then evaluate.
Pre-Calculus Mathematics 12 6.4 Sum and Difference Identities Example 5: Given angle A in quadrant I and angle B in quadrant II, such that, find ( ). Example 6: Prove: Practice: Page 292 #1-6
Pre-Calculus Mathematics 12 6.5 Double-Angle Identities Goal: 1. Identify the double angle trigonometric identities 2. Simplify and prove double-angle trigonometric expressions and equations Double Angle Identities Using the sum and difference identities, we can determine other trigonometric identities sin sin cos cos sin cos cos cos sin sin tan tan tan tan tan 1 tan tan 1 tan tan tan Double Angle Identities (same angle) sin 2 2sin cos cos 2 cos sin 2 2cos 1 2 1 2sin
Example 1: Simplify Example 2: Solve, Pre-Calculus Mathematics 12 6.5 Double-Angle Identities Example 3: Prove: Practice: Page 300 #1 6 (as many as needed)