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Angles, Degrees, and Special Triangles

Angles, Degrees, and Special Triangles. Trigonometry MATH 103 S. Rook. Overview. Section 1.1 in the textbook: Angles Degree measure Triangles Special Triangles. Angles. Angles. Angle: describes the “space” between two rays that are joined at a common endpoint

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Angles, Degrees, and Special Triangles

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  1. Angles, Degrees, and Special Triangles Trigonometry MATH 103 S. Rook

  2. Overview • Section 1.1 in the textbook: • Angles • Degree measure • Triangles • Special Triangles

  3. Angles

  4. Angles • Angle: describes the “space” between two rays that are joined at a common endpoint • Recall from Geometry that a ray has one terminating side and one non-terminating side • Can also think about an angle as a rotation about the common endpoint • Start at OA (Initial side) • End at OB (Terminal side)

  5. Angles (Continued) • If the initial side is rotated counter-clockwise θ is a positive angle • If the initial side is rotated clockwise θ is a negative angle

  6. Degree Measure

  7. Degree Measure • Degree measure: expresses the size of an angle. Often abbreviated by the symbol ° 360° makes one complete revolution • The initial and terminal sides of the angle are the same 180° makes one half of a complete revolution 90° makes one quarter of a complete revolution

  8. Degree Measure (Continued) • Angles that measure: • Between 0° and 90° are known as acute angles • Exactly 90° are known as right angles • Denoted by a small square between the initial and terminal sides • Between 90° and 180° are known as obtuse angles • Complementary angles: two angles whose measures sum to 90° • Supplementary angles: two angles whose measures sum to 180°

  9. Degree Measure (Example) Ex 1: (i) Indicate whether the angle is acute, right, or obtuse (ii) find its complement (iii) find its supplement a) 50° b) 160°

  10. Triangles

  11. Triangles • Triangle: a polygon comprised of three sides and three angles the sum of which add to 180° • The longest side is opposite the largest angle measure and the smallest side is opposite the smallest angle measure • Important types of triangles: • Equilateral: all three sides are of equal length and all three angles are of equal measure • Isosceles: two of the sides are of equal length and two of the angles are of equal measure • Scalene: all sides have a different length and all angles have a different measure

  12. Triangles (Continued) • Triangles can also be classified based on the measurement of their angles: • Acute triangle: all angles of the triangle are acute • Obtuse triangle: one angle of the triangle is obtuse • Right triangle: one angle of the triangle is a right angle • VERY important

  13. Special Triangles – Right Triangle • Pythagorean Theorem:a2 + b2 = c2 where a and b are the legs of the triangle and c is the hypotenuse • The legs are the shorter sides of the triangle • The hypotenuse is the longest side of the triangle and is opposite the 90° angle • Can be used when we have information regarding at least two sides of the triangle • The Pythagorean Theorem can ONLY be used with a RIGHT triangle

  14. Special Triangles – Right Triangle (Example) Ex 2: Find the length of the missing side: a) b) If a = 2 and c = 6, find b

  15. Special Triangles – 30° - 60° - 90° Triangle • Think about taking half of an equilateral triangle • Shortest side is x and is opposite the 30° angle • Medium side is and is opposite the 60° angle • Longest side is 2x and is opposite the 90° angle

  16. Special Triangles – 30° - 60° - 90° Triangle (Example) Ex 3: Find the length of the remaining sides: a) b) The side opposite 60° is 4

  17. Special Triangles – 45° - 45° - 90° • Think about taking half of a square along its diagonal • Shortest sides are x and are opposite the 45° angles • Longest side is and is opposite the 90° angle

  18. Special Triangles – 45° - 45° - 90° Triangle (Example) Ex 4: Find the length of the remaining sides: a) b) The longest side is

  19. Summary • After studying these slides, you should be able to: • Understand angles and angle measurement • Identify the complement or supplement of an angle • Find the third side of a right triangle when given two sides • Find the length of any side of a 30°-60°-90° triangle given the length of one of its sides • Find the length of any side of a 45°-45°-90° triangle given the length of one of its sides • Additional Practice • See the list of suggested problems for 1.1 • Next lesson • The Rectangular Coordinate System (Section 1.2)

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