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4.3 – Right Triangle Trigonometry Accelerated Pre-Calculus Mr. Niedert Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 1 / 22 4.3 – Right Triangle Trigonometry 1 The Six Trigonometric Functions Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 2 / 22 4.3 – Right Triangle Trigonometry 1 The Six Trigonometric Functions 2 Trigonometric Identities Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 2 / 22 4.3 – Right Triangle Trigonometry 1 The Six Trigonometric Functions 2 Trigonometric Identities 3 Evaluating Trigonometric Functions with a Calculator Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 2 / 22 4.3 – Right Triangle Trigonometry 1 The Six Trigonometric Functions 2 Trigonometric Identities 3 Evaluating Trigonometric Functions with a Calculator 4 Applications Involving Right Triangles Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 2 / 22 Today’s Learning Target(s) 1 I can find the values of all six trigonometric functions for any right triangle. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 3 / 22 The Six Trigonometric Functions Consider the triangle below for the definitions of the six trigonometric functions. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 4 / 22 The Six Trigonometric Functions Consider the triangle below for the definitions of the six trigonometric functions. Right Triangle Definitions of Trigonometric Functions opp hyp hyp csc θ = opp sin θ = Accelerated Pre-Calculus adj hyp hyp sec θ = adj cos θ = 4.3 – Right Triangle Trigonometry opp adj adj cot θ = opp tan θ = Mr. Niedert 4 / 22 Evaluating Trigonometric Functions Practice In the triangle below, find the values of the six trigonometric functions of θ. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 5 / 22 Complementary Angles There is a relationship between cofunctions that will help you in evaluating trigonometric functions. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 6 / 22 Complementary Angles There is a relationship between cofunctions that will help you in evaluating trigonometric functions. Sine and cosine are said to be cofunctions. Tangent and cotangent are cofunctions. In addition, secant and cosecant are cofunctions. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 6 / 22 Complementary Angles There is a relationship between cofunctions that will help you in evaluating trigonometric functions. Sine and cosine are said to be cofunctions. Tangent and cotangent are cofunctions. In addition, secant and cosecant are cofunctions. This yields the following relationships between the angles. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 6 / 22 Complementary Angles There is a relationship between cofunctions that will help you in evaluating trigonometric functions. Sine and cosine are said to be cofunctions. Tangent and cotangent are cofunctions. In addition, secant and cosecant are cofunctions. This yields the following relationships between the angles. Cofunctions are Complementary sin(90◦ − θ) = cos θ cos(90◦ − θ) = sin θ Accelerated Pre-Calculus tan(90◦ − θ) = cot θ cot(90◦ − θ) = tan θ 4.3 – Right Triangle Trigonometry sec(90◦ − θ) = csc θ csc(90◦ − θ) = sec θ Mr. Niedert 6 / 22 4.3 – Right Triangle Trigonometry (Part 1 of 3) Assignment Part 1: pg. 308 #1-4, 6-26 even Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 7 / 22 Today’s Learning Target(s) 1 I can apply the trigonometric identities to find the values of various trigonometric functions. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 8 / 22 Fundamental Trigonometric Identities Fundamental Trigonometric Identities Reciprocal Identities sin θ = 1 csc θ cos θ = 1 sec θ tan θ = 1 cot θ csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ Quotient Identities tan θ = sin θ cos θ cot θ = cos θ sin θ Pythagorean Identities sin2 θ + cos2 θ = 1 Accelerated Pre-Calculus 1 + tan2 θ = sec2 θ 1 + cot2 θ = csc2 θ 4.3 – Right Triangle Trigonometry Mr. Niedert 9 / 22 Applying Trigonometric Identities Example Let θ be an acute angle such that sin θ = 0.6. Find the values of (a) cos θ and (b) tan θ using trigonometric identities. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 10 / 22 Applying Trigonometric Identities Practice Let θ be an acute angle such that cos θ = 0.96. Find the values of (a) sin θ and (b) tan θ using trigonometric identities. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 11 / 22 Applying Trigonometric Identities Example Let θ be an acute angle such that tan θ = 3. Find the values of (a) cot θ and (b) sec θ using trigonometric identities. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 12 / 22 Applying Trigonometric Identities Practice Let β be an acute angle such that tan β = 4. Find the values of (a) cot β and (b) sec β using trigonometric identities. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 13 / 22 Transforming Trigonometric Identities Practice Use trigonometric identities to transform the left side of the equation into the right side. a tan θ cot θ = 1 b (1 + cos θ) (1 − cos θ) = sin2 θ Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 14 / 22 4.3 – Right Triangle Trigonometry (Part 2 of 3) Assignment Part 1: pg. 308 #1-4, 6-26 even Part 2: pg. 309 #28-40 even, 41-42 Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 15 / 22 Today’s Learning Target(s) 1 I can evaluate trigonometric functions using a calculator in both radians and degrees. 2 I can apply trigonometric functions to solve angles of elevation and angle of depression problems. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 16 / 22 Using a Calculator Example Use a calculator to evaluate each of the following. a cos 28◦ b sec 32◦ 150 3200 c tan 5π 12 d cot 1 Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 17 / 22 Using a Calculator Practice Use a calculator to evaluate each of the following. a sin 49◦ b csc 72◦ 350 4900 c cos 1π 5 d sec 1 Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 18 / 22 Using Trigonometry to Solve a Right Triangle Practice A surveyor is standing 115 feet from the base of the Washington Monument, as shown below. The surveyor measure the angle of elevation to the top of the monument as 78.3◦ . How tall is the Washington Monument? Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 19 / 22 Using Trigonometry to Solve a Right Triangle Practice A historic lighthouse is 200 yards from a bike path along the edge of a lake. A walkway to the lighthouse is 400 yards long. Find the acute angle θ between the bike path and the walkway. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 20 / 22 Solving a Right Triangle Practice Find the length c of the skateboard ramp below. Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 21 / 22 4.3 – Right Triangle Trigonometry (Part 3 of 3) Assignment Part 1: pg. 308 #1-4, 6-26 even Part 2: pg. 309 #28-40 even, 41-42 Part 3: pg. 309-310 #44-52 even, 59-68 4.3 – Right Triangle Trigonometry Assignment pg. 308-310 #1-4, 6-26 even, 28-40 even, 41-42, 44-52 even, 59-68 Accelerated Pre-Calculus 4.3 – Right Triangle Trigonometry Mr. Niedert 22 / 22
