class notes.notebook 1 December 04, 2013 Chapter 4 – Trigonometry & the Unit Circle 4.1 Angles & Angle Measure
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Chapter 4 – Trigonometry & the Unit Circle
4.1 Angles & Angle Measure
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By converntion, angles measured on clockwise direction are positive while those in a counterclockwise are said to be negative.
****Note: Angles measures without units are considered to be radians
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How many degrees are in 1 radian?
These two formulae can now be used to convert angular measure into degrees or radians.
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Draw each angle in standard position. Change each degree measure to radian measure and each radian measure to degree measure. Give answers as both exact and approximate measures (if necessary) to the nearest hundredth of a unit.
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a) b) c)
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a)
b) c)
d)
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Coterminal Angles
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Example 3 (pg 172): Express the angles coterminal with 110 degrees in general form. Identify the angles that satisfy the domain 7200 ≤ x < 7200
Ex 3B: Express the angles coterminal with in general form. Identify the angles that satisfy the domain
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All arcs that subtend a right angle ( ) have the same central angle, but they have different arc lengths depending on the radius of the circle. The arc length is proportional to the radius. This is true for any central angle and related arc length.
Consider the circle with radius r and the sector with central angle .
This formula, a = r, works for any circle, where is in radians.
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Rosemarie is taking a course in industrial engineering. For anassignment, she is designing the interface of a DVD player. In her plan, she includes a decorative arc below the on/off button. The arc has central angle 130° in a circle with radius 6.7 mm. Determine the length of the arc, to the nearest tenth of a mm.
Example 4:
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A circle with centre at the origin (0,0) and a radius of 1 unit is referred to as a unit circle.
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(a) Identify the radius of the circle.
(b) Identify the coordinates of the point P.
(c) Draw a vertical line segment PN from P to the x-axis.
(d) Draw a horizontal line segment NO from N to the origin.
(e) What type of triangle have you created?
(f) Determine the lengths of PN and NO.
(g) Write an equation that relates the three sides of the triangle.
(h) Write an equation of any circle that is centered at the origin and has a radius of ‘r’.
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a) radius = 6 b) radius = c) diameter = 6d) passing through
the point (-5, 12)
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Determine the missing coordinate(s) for all points on the unit circle satisfying the given conditions. Use the circle diagram and determine which quadrant(s) the points lie in.
where the point is in quadrant II
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The formin radians; and r is the radius, applies to any circle, as long as a and r are measured in the same units. In the unit circle, the formula becomes a = θ(1) or a = θ. This means that a central angle and its subtended arc on the unit circle have the same numerical value.
You can use the function P(θ) = (x, y) to link the arc length, θ, of acentral angle in the unit circle to the coordinates, (x, y), of the point of intersection of the terminal arm and the unit circle.
For example, ifθ = π, the point is (-1, 0). Thus, youcan write P(π) = (-1, 0).
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4.3 Trigonometric Ratios
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Reciprocal Trigonometric RatiosCosecant Secant Cotangent
Note:
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