Download 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
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Angles
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)
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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
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Degree Measure
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
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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°
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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°
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Triangles
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
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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
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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
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Special Triangles – Right Triangle
(Example)
Ex 2: Find the length of the missing side:
a)
b) If a = 2 and c = 6, find b
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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 x 3 and is opposite the 60°
angle
– Longest side is 2x and is
opposite the 90° angle
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Special Triangles – 30° - 60° - 90°
Triangle (Example)
Ex 3: Find the length of the remaining sides:
a)
b) The side opposite 60° is 4
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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 x 2 and is
opposite the 90° angle
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Special Triangles – 45° - 45° - 90°
Triangle (Example)
Ex 4: Find the length of the remaining sides:
a)
b) The longest side is 5 2
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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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