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Introduction to Trigonometry

Introduction to Trigonometry. Basic Trigonometric Functions. What is Trigonometry?. The study of triangles Relationship between sides and angles of a right triangle What is a right triangle? A triangle with a 90 ⁰ angle. 90°. Review Right Triangles.

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Introduction to Trigonometry

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  1. Introduction to Trigonometry Basic Trigonometric Functions

  2. What is Trigonometry? • The study of triangles • Relationship between sides and angles of a right triangle • What is a right triangle? A triangle with a 90⁰ angle 90°

  3. Review Right Triangles • In relation to angle a, the sides of the triangle are: • hypotenuse- always longest side and side across from right angle (90⁰) • adjacent - side closest to angle a • opposite - side opposite toangle a a hypotenuse adjacent 90⁰ opposite

  4. Review Right Triangles Label the sides for angle b: • hypotenuse • adjacent • opposite ? hypotenuse ? opposite 90° b ? adjacent

  5. Trigonometric Functions Ratios of the sides in relation to angle a: • sine • cosine • tangent a hypotenuse adjacent 90° opposite

  6. Trigonometric Functions:SINE abbreviation: sin sin(a)= • 0 ≤ sin ≤ 1 • Example: sin(60°)= = ~.866 • Ratio of opposite side to hypotenuse for 60° angle is to 2 (.866 to 1) opposite hypotenuse a hypotenuse 2 adjacent 90° opposite

  7. Trigonometric Functions:COSINE abbreviation: cos cos(a)= • 0 ≤ cos ≤ 1 • Example: cos(60°)= = .5 • Ratio of adjacent side to hypotenuse for 60° angle is 1 to 2 (half) adjacent hypotenuse a 1 hypotenuse 2 adjacent 90° opposite

  8. Trigonometric Functions:TANGENT abbreviation: tan tan(a)= • 0 ≤ tan ≤ • Example: tan(60°)= = ~1.732 • Ratio of opposite side to adjacent side for 60° angle is to 1 (1.732 to 1) opposite opposite adjacent adjacent a ∞ hypotenuse adjacent 90° opposite

  9. Trigonometric Functions: SOH SOH – CAH – TOA CAH TOA REMEMBER: Sine = Cosine = Tangent = Opposite Hypotenuse a Adjacent Hypotenuse hypotenuse Opposite adjacent Adjacent 90° opposite

  10. Using Trigonometric Functions: For any right triangle: • calculate other sides if one side and angle known • calculate angle if two sides known 90°

  11. Calculating Sides:One Side and Angle Known What is known? • angle (50°) and adjacent side (2) Solving for hypotenuse: Which function uses adjacent and hypotenuse? 50° hypotenuse 2 COSINE 90° opposite

  12. Calculating Sides:One Side and Angle Known What is known? • angle (50°) and adjacent side (2) Solving for hypotenuse: cos(50°)= = hypotenuse = ~3.111 50° 2 ~0.643 3.111 hypotenuse hypotenuse 2 90° opposite

  13. Calculating Sides:One Side and Angle Known Now we know: • angle (50°) and hypotenuse (3.111) Solving for opposite: Which function uses oppositeand hypotenuse? 50° 3.111 2 SINE 90° opposite

  14. Calculating Sides:One Side and Angle Known Now we know: • angle (50°) and hypotenuse(3.111) Solving for opposite: sin(50°)= = opposite= ~2.384 50° opposite ~.766 3.111 3.111 2 90° 2.384 opposite

  15. Calculating Angle:Two Sides Known What is known? • adjacent (3) and opposite (5) Solving for angle (a): Which function uses adjacent and opposite? a hypotenuse 3 TANGENT 90° 5

  16. Calculating Angle:Two Sides Known What is known? • adjacent (3) and opposite (5) Solving for angle (a): tan(a)= = * need to use inverse tan → tan-1(.6) = a= ~30.964° a 30.964° 3 .6 5 hypotenuse 3 90° 5

  17. TEKS Reference §111.35. Precalculus (c)  Knowledge and skills. (3)  The student uses functions and their properties, tools and technology, to model and solve meaningful problems. The student is expected to: (A)  investigate properties of trigonometric and polynomial functions;

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