Right Triangle Trigonometry 23 March 2011

Degree Mode v. Radian Mode

Symbols Theda – Represents the angle measure Hypotenuse Opposite Side Adjacent Side

Six Trigonometric Ratios 3 Basic Ratios + 3 Reciprocal Ratios What is a reciprocal?

Six Trigonometric Ratios, cont. Basic Trig. Ratio Sine Cosine Tangent Reciprocal Trig. Ratio Cosecant Secant Cotangent It’s a sin to have two c’s.

Three Basic Trig. Ratios SOH-CAH-TOA

Sine (SOH)

Cosine (CAH)

Tangent (TOA)

Trigonometric Functions, cont. 3 Reciprocal Functions Cosecant – Reciprocal of Sine Secant – Reciprocal of Cosine Cotangent – Reciprocal of Tangent Remember, “It’s a sin to have two C’s”

Cosecant – Reciprocal of Sine (“It’s a sin to have two C’s.”)

Secant – Reciprocal of Cosine

Cotangent – Reciprocal of Tangent

Your Turn: Pg. 419: 9 – 14, 27 – 32

Solving for Side Lengths If given one side and one angle measure, then we can solve for any other side of the triangle. 8 x

Solving Right Triangles, cont. 1. Ask yourself what types of sides do you have: opposite, adjacent, and/or hypotenuse? 2. Pick the appropriate trig function to solve for x. 3. Solve for x. 8 x

Solving for Side Lengths, cont. 8 x

Solving Side Lengths, cont. 14x

Special Trigonometric Ratios Memorize These!!! 30°45°60° sin cos tan

Your Turn:

Inverse Trigonometric Ratios We can “undo” trig ratios Gives us the angle measurement (theda) Represented by a small –1 in the upper right hand corner Ex. 2 nd button → correct trig ratio

Inverse Trigonometric Ratios, cont.

Your Turn: Solve for theda Round to nearest hundredth

Solving For Angle Measures If given two sides of a triangle, then we can solve for any of the angles of the triangle. 54

Solving for Angle Measures, cont. 1. Ask yourself what types of sides do you have: opposite, adjacent, and/or hypotenuse? 2. Pick the appropriate trig function to solve for 3. Solve for using the inverse trigonometric function 54

Solving for Angle Measures, cont. 54

Your Turn: Complete problems 11 – 16 on the Solving Right Triangles Practice handout

Solving Right Triangles We can use two properties of triangles to solve for all the angles and the side lengths of a right triangle.

Properties of Triangles Pythagorean Theorem For a right triangle, a 2 + b 2 = c 2 Triangle Sum Theorem When you add up all the angles in a triangle, they equal 180°

Tricks for Solving Right Triangles Given Two Sides 1. Use Pythagorean Theorem to solve for remaining side. 2. Solve for 1 of the angles using trig ratios 3. Solve for the other angle using Triangle Sum Theorem Given an Angle & a Side 1. Use the Triangle Sum Theorem to solve for the other angle 2. Use trig ratios to solve for 1 of the sides 3. Use the Pythagorean Theorem to solve for the other side

Beta – Another symbol for an unknown angle measure

Solving Right Triangles – Examples: Given Two Sides 54

Solving Right Triangles – Examples: Given an Angle and a Side 2 30°