 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°

 In relation to angle a, the sides of the triangle are: hypotenuse - always longest side and side across from right angle ( 90 ⁰ ) adjacent - side closest to angle a opposite - side opposite to angle a hypotenuse a adjacent opposite 90 ⁰

Label the sides for angle b : hypotenuse adjacent opposite b ? ? ? hypotenuse opposite adjacent 90°

Ratios of the sides in relation to angle a :  sine  cosine  tangent hypotenuse a adjacent opposite 90°

hypotenuse a adjacent opposite 90° opposite hypotenuse 2

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) h y p o t e n u s e a a d j a c e n t opposite 90° adjacent hypotenuse 1 2

a adjacent opposite 90° opposite adjacent ∞ opposite adjacent

REMEMBER: S ine = C osine = T angent = hypotenuse a adjacent opposite 90° O pposite A djacent H ypotenuse A djacent O pposite SOH – CAH – TOA SOHTOACAH

For any right triangle:  calculate other sides if one side and angle known  calculate angle if two sides known 90°

What is known? angle (50°) and adjacent side (2) Solving for hypotenuse : Which function uses adjacent and hypotenuse? hypotenuse 50° 2 opposite 90° COSINE

What is known? angle (50°) and adjacent side (2) Solving for hypotenuse : cos(50°)= = hypotenuse = ~3.111 hypotenuse 50° 2 opposite 90° 2 hypotenuse ~0.643

Now we know: angle (50°) and hypotenuse (3.111) Solving for opposite : Which function uses opposite and hypotenuse? ° 2 opposite 90° SINE

Now we know: angle (50°) and hypotenuse (3.111) Solving for opposite : sin(50°)= = opposite = ~ ° 2 opposite 90° opposite ~.766

What is known? adjacent (3) and opposite (5) Solving for angle (a) : Which function uses adjacent and opposite? hypotenuse a ° TANGENT

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° hypotenuse ° 3 5 a °.6

§ 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;