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Fundamental Trigonometric Identities by CHED on May 07, 2017 lesson duration of 41 minutes under Precalculus generated on May 07, 2017 at 04:52 am Tags: Trigonometry CHED.GOV.PH K-12 Teacher's Resource Community Generated: May 07,2017 12:52 PM Fundamental Trigonometric Identities ( 1 hour and 41 mins ) Written By: CHED on July 4, 2016 Subjects: Precalculus Tags: Trigonometry Resources N/A N/A Content Standard Key concepts of circular functions, trigonometric identities, inverse trigonometric functions, and the polar coordinate system Performance Standard Formulate and solve accurately situational problems involving circular functions Apply appropriate trigonometric identities in solving situational problems Formulate and solve accurately situational problems involving appropriate trigonometric functions Formulate and solve accurately situational problems involving the polar coordinate system Learning Competencies Illustrate the different circular functions Introduction 1 mins In previous lessons, we have defined trigonometric functions using the unit circle and also investigated the graphs of the six trigonometric functions. This lesson builds on the understanding of the different trigonometric functions by discovery, deriving, and working with trigonometric identities. Domain of an Expression or Equation Consider the following expressions: What are the real values of the variable x that make the expressions defined in the set of real numbers? 1 / 15 CHED.GOV.PH K-12 Teacher's Resource Community thefirst firstexpression, expression,every everyreal realvalue valueofof x when whensubstituted substitutedtotothe theexpression expressionmakes makesit itdefined definedininthe theset setofofreal real InInthe numbers; that is, the value of the expression is real when x is real. Inthe thesecond secondexpression, expression,not notevery everyreal realvalue valueof ofxxmakes makesthe theexpression expressiondefined definedin in R. For example, when x = 0, the In expression becomes square root of ?1, which is not a real number. Here, for square root of (x^2 ? 1) to be defined in R, x must be in (?infinity,?1]U[1,infinity). In the third expression, the values of x that make the denominator zero make the entire expression undefined. x^2 ? 3x 3x ? 4 = (x (x ? 4)(x 4)(x + 1) = 0 <=> x = 4 or x = ?1 Hence, the expression x/(x^2 ? 3x ? 4) is real when x =/= 4 and x =/= ?1. In the fourth expression, because the expression square root of (x (x ? 1) is in the denominator, x must be greater than 1. Although the value of the entire expression is 0 when x = 0, we do not include 0 as allowed value of x because part of the expression is not real when x = 0. In the expressions above, the allowed values of the variable x constitute the domain of the expression. In the expressions above, the domains of the first, second, third, and fourth expressions are R, (?infinity,?1] U [1,infinity), R \ {?1, 4}, and (1,infinity), respectively. Example 3.4.1. Determine the domain of the expression/equation. 2 / 15 CHED.GOV.PH K-12 Teacher's Resource Community Identity and Conditional Equation Consider the following two groups of equations: In each equation in Group A, some values of the variable that are in the domain of the equation do not satisfy the equation (that is, do not make the equation true). On the other hand, in each equation in Group B, every element in the domain of the equation satisfies the given equation. The equations in Group A are called conditional equations, while those in Group B are called identities. 3 / 15 CHED.GOV.PH K-12 Teacher's Resource Community Example 3.4.2. Identify Identify whether given equation is an identity a conditional equation. equation. For For each each conditional conditional whether thethe given equation is an identity or or a conditional equation, provide a value of the variable in the domain that does not satisfy the equation. (1) x^3 ?2 = (x ? cube root of 2) (x^2 + cube root of 2x + cube root of 4) (2) sin ^2 ? = cos^2 + 1 (3) sin ? = cos ? ? 1 (4) (1 ? square root of x)/(1 + square root of x) = (1 ? (2 square root of x + x))/(1 ? x) Solution. (1) This is an identity because this is simply factoring of difference of two cubes. (2) This is a conditional equation. If ? = 0, then the left-hand side of the equation is 0, while the right-hand side is 2. (3) This is also a conditional equation. If ? = 0, then both sides of the equation are equal to 0. But if ? = pi, then the lefthand side of the equation is 0, while the right-hand side is ?2. (4) This is an identity because the right-hand side of the equation is obtained by rationalizing the denominator of the left-hand side. The Fundamental Trigonometric Identities Recall that if P(x, y) is the terminal point on the unit circle corresponding to ?, then we have From the definitions, the following reciprocal and quotient identities immediately follow. Note that these identities hold if ? is taken either as a real number or as an angle. 4 / 15 CHED.GOV.PH K-12 Teacher's Resource Community We can use these identities to simplify trigonometric expressions. Example 3.4.3. Simplify: If P(x, y) is the terminal point on the unit circle corresponding to ?, then x^2 + y^2 = 1. Since sin ? = y and cos ? = x, we get sin^2 ? + cos^2 ? = 1. By dividing both sides of this identity by cos^2 ? and sin^2 ?, respectively, we obtain tan2^ ? + 1 = sec2^ ? and 1 + cot^2 ? = csc^2 ?. Teaching Notes The assumption in the division is that the divisor is nonzero. 5 / 15 CHED.GOV.PH K-12 Teacher's Resource Community In addition to the eight identities presented above, we also have the following identities. The first two of the negative identities can be obtained from the graphs of the sine and cosine functions, respectively. (Please review the discussion on page 145.) The third identity can be derived as follows: Teaching Notes The corresponding reciprocal functions follow the same Even-Odd Identities: csc(??) = ?csc ? sec(??) = sec? cot(??) = ?cot ?. The reciprocal, quotient, Pythagorean, and even-odd identities constitute what we call the fundamental trigonometric identities. identities. We now solve Example 3.2.3 in a different way. Example 3.4.5. If sin ? = ?3/4 and cos ? > 0. Find cos ?. Solution. Using the identity sin^2 ? + cos^2 ? = 1 with cos ? > 0, we have 6 / 15 CHED.GOV.PH K-12 Teacher's Resource Community Example 3.4.6. If sec ? = 5/2 and tan ? < 0, use the identities to find the values of the remaining trigonometric functions of ?. Solution. Note that ? lies in QIV. Proving Trigonometric Identities We can use the eleven fundamental trigonometric identities to establish other identities. For example, suppose we want to establish the identity csc ? ? cot ? = sin ?/(1 + cos ?) To verify that it is an identity, recall that we need to establish the truth of the equation for all values of the variable in the domain of the equation. It is not enough to verify its truth for some selected values of the variable. To prove it, we use the fundamental trigonometric identities and valid algebraic manipulations like performing the fundamental operations, factoring, canceling, and multiplying the numerator and denominator by the same quantity. Start on the expression on one side of the proposed identity (preferably the complicated side), use and apply some of the fundamental trigonometric identities and algebraic manipulations, and arrive at the expression on the other side of the proposed identity. Upon arriving at the expression of the other side, the identity has been established. There is no unique technique to 7 / 15 CHED.GOV.PH K-12 Teacher's Resource Community prove all identities, but familiarity with the different techniques may help. Example 3.4.7. Prove: sec x ? cos x = sin x tan x. Solution. Seatwork 0 mins Find the domain of the expression/equation. Seatwork/Homework 3.4.2 0 mins Seatwork/Homework 3.4.2 8 / 15 CHED.GOV.PH K-12 Teacher's Resource Community Identify whether the given equation is an identity or a conditional equation. For each conditional equation, provide a value of the variable in the domain that does not satisfy the equation. Seatwork/Homework 3.4.3 99 mins 1. Use the identities presented in this lesson to simplify each trigonometric expression. 9 / 15 CHED.GOV.PH K-12 Teacher's Resource Community 2. Given some initial values, use the identities to find the values of the remaining trigonometric functions of ?. Seatwork/Homework 3.4.4 0 mins Prove each identity. 1. tan x + cot x = csc x sec x 10 / 15 CHED.GOV.PH K-12 Teacher's Resource Community 2. sec ? + tan ? = 1/(sec ? ? tan ?) Answer: 2 csc ? 3. sec y + tan y csc y + 1 = tan y 4. 2 csc^2 ? = 1/(1 ? cos ?) + 1/(1 + cos ?) 11 / 15 CHED.GOV.PH K-12 Teacher's Resource Community Exercise 1 mins 1. Find the domain of the equation. 2. Simplify each expression using the fundamental identities. 3. Given some initial information, use the identities to find the values of the trigonometric functions of ?. 12 / 15 CHED.GOV.PH K-12 Teacher's Resource Community 4. Determine whether the given equation is an identity or a conditional equation. If it is an identity, prove it; otherwise, provide a value of the variable in the domain that does not satisfy the equation. 13 / 15 CHED.GOV.PH K-12 Teacher's Resource Community 5. Prove the following identities. 6. Express (1 ? sec^2 x)/sec^2 x in terms of sin x. Answer: ?sin^2 x 7. Express tan x sec x in terms of cos x. 14 / 15 CHED.GOV.PH K-12 Teacher's Resource Community 8. Express all other five trigonometric functions in terms of tan x (allowing ± in the expression). 9. If sec ? ? tan ? = 3, what is sec ? + tan ?? Answer. 1/3 Solution. tan^2 ? + 1 = sec^2 ? sec^2 ? ? tan^2 ? = 1 (sec ? ? tan ?)(sec ? + tan ?) = 1 3(sec ? + tan?) = 1 sec ? + tan ? = 1/3 Generated: May 07,2017 12:52 PM 15 / 15 Powered Poweredby byTCPDF TCPDF(www.tcpdf.org) (www.tcpdf.org)