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Right Triangles & Trigonometry

This resource delves into the key concepts of right triangles and trigonometry. It covers vital objectives such as finding geometric means, employing the Pythagorean Theorem, and understanding special right triangles (45°-45°-90° and 30°-60°-90°). Explore how the altitude from the right angle creates two similar triangles and functions as a geometric mean. Additionally, learn about trigonometric ratios (sine, cosine, tangent) and their applications in solving real-world problems, including angles of elevation and depression.

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Right Triangles & Trigonometry

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  1. Right Triangles & Trigonometry OBJECTIVES: Using Geometric mean Pythagorean Theorem 45°- 45°- 90° and 30°-60°-90° rt. Δ’s trig in solving Δ’s

  2. A C Geometric Means D B Finding the geometric mean between a & b: • The altitude from the 90°angle of a rt. Δ forms 2 similar rt. Δ’s • This altitude is the geometric mean between the 2 hypotenuses and forms the proportion:

  3. The Pythagorean Theorem The sum of the square of legs = hypotenuse 2 • A Pythagorean Triple is a group of 3 numbers that ‘fits’ the Pythagorean Theorem such as 3-4-5, 5-12-13, 7-24-25, 8-15-17,… a2 + b2 = c2 c a b

  4. Special Right Triangles In a 45°- 45°- 90° Δ, the hypotenuse = In a 30°-60°-90° Δ, the hyp = 2 • shorter leg the longer leg = shorter leg ( the shorter leg is opposite the 30 ° angle & the longer leg is opposite the 60° angle) x x 30° 2x x

  5. B Trig Ratios in Right Triangles hypotenuse opp / A c a A b C adjacent to / A Trig Ratio Definition Sine of / A = sin A = Cosine / A = cos A = Tangent / A = tan A = Trig ratios use the acute angles of a rt. Δ. Start with / & find the side opp & the side adj.

  6. Using trig: Angles of elevation & depression Angle of elevation • Angles of elevation & depression are measured from a line // to horizon ( looking up or down) • Use trig ratios to solve for distances, height, etc ~~~~~~~~~~~~~~~~~~~~~~~

  7. Law of Sines B c a To solve any triangle (not just a rt. Δ), use the Law of Sines to help find the 3 angles & the 3 sides Given: B = 39° & C = 88 °, b = 10 A b C Use the first two fractions & cross mult for a; then use the 2nd & 3rd fractions for c

  8. Use Law of Cosines with any Δ C b a A c B • Use law of Cosines once when the info doesn’t fit for the law of Sines, then use law of Sines • Use the formula for the angle you’re given. Be careful!

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