Chapter 6 Trigonometry 6 1 RightTriangle Trigonometry Essential

Chapter 6: Trigonometry 6. 1: Right-Triangle Trigonometry Essential Question: What are the six trigonometric functions and how are they related?

6. 1: Right-Triangle Trigonometry • Angles and Degree Measure – An angle is formed by two rays with a common endpoint – That endpoint is called the vertex – Angles can be labeled by the angle symbol ( ) and the vertex. The angle below may be labeled A side vertex angle A side

6. 1: Right-Triangle Trigonometry Angles › Measured in degrees 1 degree (°) is 1/360 of a circle 180° is ½ a circle 90° (a right angle) is ¼ a circle Minutes and seconds › A minute (‘) is 1/60 of a degree › A second (“) is 1/60 of a minute Hence, 1 second = 1/3600 of a degree Hint: It may be easier to convert between degree and DMS (Degree – Minute – Second) forms if you think of degrees as hours

6. 1: Right-Triangle Trigonometry • Converting between decimal form and DMS form • Write 35° 15’ 27” in decimal form – Divide minutes by 60 – Divide seconds by 3600 – Add all numbers together –

6. 1: Right-Triangle Trigonometry • Converting between decimal form and DMS form (part 2) • Write 48. 3625° in DMS form – The whole number is the degree. – Multiply by 60. The whole number is the minute. – Multiply by 60. The whole number is the second –

6. 1: Right-Triangle Trigonometry • Similar Triangles and Trigonometric Ratios – Pneumonic Phrase: SOH-CAH-TOA – Learn it, live it, love it • sin (sine) = opposite / hypotenuse • cos (cosine) = adjacent / hypotenuse • tan (tangent) = opposite / adjacent – Reciprocals of trigonometric ratios • csc (cosecant) = 1 / sin = hypotenuse / opposite • sec (secant) = 1 / cos = hypotenuse / adjacent • cot (cotangent) = 1 / tan = adjacent / opposite

6. 1: Right-Triangle Trigonometry • Evaluating Trigonometric Ratios – Evaluate the six trigonometric ratios of the angle θ shown below – 13 5 θ 12 • sin θ = 5 / 13 • cos θ = 12 / 13 • tan θ = 5 / 12 csc θ = 13 / 5 sec θ = 13 / 12 cot θ = 12 / 5

6. 1: Right-Triangle Trigonometry • Evaluating Trigonometric Ratios – Evaluate the six trigonometric ratios of 62° by using the triangle shown below 3 – 1. 6 62° θ 3. 4 • sin 62° = 3 / 3. 4 csc 62° = 3. 4 / 3 • cos 62° = 1. 6 / 3. 4 sec 62° = 3. 4 / 1. 6 • tan 62° = 3 / 1. 6 cot 62° = 1. 6 / 3

6. 1: Right-Triangle Trigonometry Evaluating Trigonometric Ratios on a calculator › Calculators have buttons to evaluate sin, cos, tan › To get the trigonometric ratio, simply input the angle value into your calculator. Make sure you are in degree mode 2 nd → MORE → 3 rd item down (we’ll be using radians later) › Evaluate the six trigonometric ratios of 20° sin 20° ≈ 0. 3420 cos 20° ≈ 0. 9397 tan 20° ≈ 0. 3640 csc 20° = 1 / sin 20° ≈ 2. 9238 sec 20° = 1 / cos 20° ≈ 1. 0642 cot 20° = 1 / tan 20° ≈ 2. 7475

6. 1: Right-Triangle Trigonometry • Special Angles – For 30 -60 -90 and 45 -45 -90 triangles, we use exact values instead of calculator estimates. – 2 60° 45° 1 1 30° 45° 1

6. 1: Right-Triangle Trigonometry • Assignment – Page 419 • Problems 9 – 31, odd problems • Copy the chart at the top of page 419 • Begin to memorize the terms from that chart which are highlighted in blue