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

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

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) 4

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 5

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 7

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° 8

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° 9

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 11

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 12

Special Triangles – Right Triangle Pythagorean Theorem: a 2 + b 2 = c 2 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 13

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

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 15

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

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 17

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

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) 19