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 sin θ = 5 / 13 csc θ = 13 / 5 cos θ = 12 / 13 sec θ = 13 / 12 tan θ = 5 / 12 cot θ = 12 / 5 13 5 θ 12

6.1: Right-Triangle Trigonometry Evaluating Trigonometric Ratios Evaluate the six trigonometric ratios of 62° by using the triangle shown below 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 3 1.6 θ 62° 3.4

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 2nd → More → 3rd item down (we’ll be using radians later) Evaluate the six trigonometric ratios of 20° sin 20° ≈ 0.3420 csc 20° = 1 / sin 20° ≈ 2.9238 cos 20° ≈ 0.9397 sec 20° = 1 / cos 20° ≈ 1.0642 tan 20° ≈ 0.3640 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. 60° 2 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