Unit 5: Trigonometry Final Exam Review

Topics to Include Pythagorean Theorem Converse to the Pythagorean Theorem Special Right Triangles Trigonometry Find a Missing Side Length Find a Missing Angle Measure Angle of Elevation and Depression

Pythagorean Theorem The Pythagorean Theorem is used to find MISSING SIDE LENGTHS in RIGHT triangles. The theorem: a2 + b2 = c2 a and b are the LEGS c must be the HYPOTENUSE

Pythagorean Theorem Example a2 + b2 = c2 62 + 52 = x2 36 + 25 = x2 61 =𝑥 7.8 = x

Pythagorean Theorem Now you try:

Converse to the Pythagorean Theorem You can also use the Pythagorean Theorem to determine if a triangle is a RIGHT triangle If you perform the Pythagorean Theorem and the equation is EQUAL, then the triangle IS a right triangle If you perform the Pythagorean Theorem and the equation is NOT EQUAL, then the triangle is NOT a right triangle If 𝒄 𝟐 < 𝒂 𝟐 + 𝒃 𝟐 , then the triangle is ACUTE If 𝒄 𝟐 > 𝒂 𝟐 + 𝒃 𝟐 , then the triangle is OBTUSE

Converse to the Pythagorean Theorem Example Side lengths: 4, 12, 10 a2 + b2 = c2 42 + 102 = 122 16 + 100 = 144 116 = 144 Since 144 is greater than 116, the triangle is OBTUSE

Converse to the Pythagorean Theorem Now you try: 1. Side lengths: 9, 40, 41 2. Side lengths: 8, 7, 5

Special Right Triangles There are 2 special right triangles that have side lengths that always have a common ratio. They are considered SHORT CUTS to the Pythagorean theorem. 45 – 45 – 90 30 – 60 – 90

Special Right Triangles Example 1. 2. X = 10 U = 4 Y = 10 𝟐 V = 2 𝟑

Special Right Triangles You Try 1. 2. 3. X = ________ X = ________ X = ________ Y = ________ Y = ________ Y = ________

Trigonometry Trigonometry is used in order to find missing SIDE LENGTHS and missing ANGLE MEASURES in a right triangle There are 3 trigonometric ratios SINE COSINE TANGENT In order to correctly find the missing side or angle, you must use the CORRECT ratio

Trigonometry SOHCAHTOA Sine Cosine Tangent OPPOSITE over HYPOTENUSE ADJACENT over HYPOTENUSE Tangent OPPOSITE over ADJACENT

Trigonometry Example Which trig ratio would you use? You would use TANGENT You would use SINE

Finding a Missing Side Length To find a missing side length: Determine which RATIO to use (sine, cosine, or tangent) Set up the problem using the given ANGLE measure Put a 1 under the trig ratio CROSS multiply Make sure your calculator is in DEGREE MODE

Find a Missing Side Length Watch this video for an example on how to find a missing side length

Find a Missing Side Length Now try this:

Finding a Missing Angle Measure To find a missing angle measure: Determine which RATIO to use (sine, cosine, or tangent) Set up the problem using the given ANGLE measure Use the INVERSE trig button in your calculator Make sure your calculator is in DEGREE MODE

Find a Missing Angle measure Watch this video for an example on how to find a missing angle measure

Finding a Missing Angle Measure Now you try: 1. 2.

Angle of Elevation and Depression Angles of Elevation and Depression are used in WORD PROBLEMS in order to find missing angles and side lengths Angle of Elevation Angle of Depression The Angle of Elevation and the Angle of Depression are ALWAYS EQUAL

Angle of Elevation and Depression Now you try: An escalator from the ground floor to the second floor of a department store is 110 ft long and rises 32 ft. vertically. What is the escalator’s angle of elevation? From the top of a lighthouse 210 feet high, the angle of depression of a boat is 27. Find the distance from the boat to the foot of the lighthouse. The lighthouse was built at sea level.

ALL DONE