This document provides an overview of right triangle trigonometry including definitions of sides and angles, the sine, cosine, and tangent functions, and how to use them to solve problems. It defines the opposite, adjacent, and hypotenuse sides of a right triangle in relation to the angle θ. Sine is defined as opposite/hypotenuse, cosine as adjacent/hypotenuse, and tangent as opposite/adjacent. Examples are provided to demonstrate using trig functions to find missing side lengths or angle measures.

1. Right Triangle TrigonometryHypotenuseOpposite𝜃 Adjacent

2. ReviewSides and Angles of a Right TriangleTheta= angle in a right triangle (𝜃)Opposite= side opposite of angleAdjacent= side next to angleHypotenuse= long side of right triangle Anybody need to see a picture?

3. 3 Functions for Right TrianglesSine (sin)Cosine (cos)Tangent (tan)

4. What are these Functions Used For?Shows relationships between sides and anglesFind the length of missing side:When given an angle and sideFind measure of the missing angleWhen given the length of sides

5. SineSine = Opposite/HypotenuseHypotenuseOpposite𝜃

6. CosineCos = Adjacent/HypotenuseType equation here. Hypotenuse𝜃 Adjacent

7. TangentTan = Opposite/AdjacentOpposite𝜃 Adjacent

8. Way to Remember Which Sides to DivideSOHCAHTOASOHSin = Opposite/HypotenuseCAHCos = Adjacent/HypotenuseTOATan = Opposite/Adjacent

9. Example 1I will do this one with youRecall the 45°- 45°- 90°- right triangleFind sin, cos, tan of 45° angle in the bottom right Type equation here. 45° √2 145° 90° 1

10. Example 1 continuedSin(θ) = opposite/hypotenuseSin(45°) = 1/√2 ≈ .707Cos(θ) = adjacent/hypotenuseCos(45°) = 1/√2 ≈ .707Tan(θ) = opposite/adjacentTan(45°) = 1/1 = 1

11. Example 2Work in groups of 3Find sin, cos, tan of 70° angleRound your answers to the nearest hundredth5.55.170°1.9

12. Example 2 continuedSin(θ) = opposite/hypotenuseSin(70°) = 5.1/5.5 ≈ .93Check answer using calculatorCos(θ) = adjacent/hypotenuseCos(70°) = 1.9/5.5 ≈ .35CalculatorTan(θ) = opposite/adjacentTan(70°) = 5.1/1.9 ≈ 2.69Calculator

13. What Happens When…?Find the length of xWhat trig function do we have?Sin(63°) = x/7Multiply 7 by sin(63°) to get answerUse your calculator to round to nearest tenthAnswer: x ≈ 6.27x63°

14. What about…?Find the measure of the angleRound to the nearest degreeWhat trig function is shown here?Cosine10𝜃 4

15. What about…? continuedSet up the function so it can be solvedCos (𝜃) = 4/10To solve for 𝜃:Need to get 𝜃 by itselfOn your calculators:Hit the 2nd button on your calculatorPush the cos button2nd button gives you inverse cosine (cos-1)So cos-1(4/10) is about 66 degreesAny questions?

16. What Did We Learn Today?Reviewed angles, sides of right triangleWhat 𝜃 isSin, cos, and tanWhat they areHow they are usedREMEMBER SOHCAHTOA!!Special casesDrawings and symbols used by Microsoft Arts