This module is concerned with the solution of the problems with respect to theory. Quick solutions are not available for all questions. Join us for STEP 2 next semester for discussions on Calculator Techniques!
THE UNIT CIRCLE
To convert from degrees to radians, use conversion factor: 180° = π
THE TRIG FUNCTIONS SOH sin θ = opposite/hypotenuse
CAH cos θ = adjacent/hypotenuse
TOA tan θ = opposite/adjacent RECIPROCALS csc θ = 1/sin θ sec θ = 1/cos θ cot θ = 1/tan θ
THE TRIG FUNCTIONS See Pre-Test Problem: A person 100 meters from the base of a tree, observes that the angle between the ground and the top of the tree is 18 degrees. Estimate the height of the tree.
H 18° tree
100 m
tan θ = opp/adj tan 18 = H/100
H = (100m) tan(18) H = 32.5 m
person
TRIG IDENTITIES Identities help to solve complicated expressions by converting them into ones that may be simplified or neglected altogether. Some common identities:
LINES Slopes of lines measure ratio of the change in y value against the change in x value between two points. Generally, đ?‘Œ −đ?‘Œ
slope (m) = đ?‘‹2−đ?‘‹1 . 2
1
Lines are expressed most commonly in the point-slope form:
y = mx + b Relationships between lines: Two lines may be called parallel or perpendicular. - Lines are parallel when they have the same slope (m) but different yintercept (b) - Lines are perpendicular when the slope of one line is the negative reciprocal of the other, m1 = -1/m2.
LINES See Pre-Test problem: What is the slope of a line perpendicular to a second line passing through (5,2) and (9,1)? Slope of points:
Δ� Δ�
→
1−2 9−5
=−
Perpendicularity → m’ = − m’ = −
1 −
1 4
=4
1 đ?‘š
1 4
CONICS: Circles Standard equation for circles:
(x - h)2 + (y - k)2 = r2 Circles at the origin have h and k equal to zero since (h, k) is the center of the circle.
CONICS: parabolas There are two types of parabolas: those that open vertically (either downward or upward) or horizontally (to the sides, either to the left or right).
General equation for parabolas:
(x - h)2 = Âą4c (y - k)
(y -
Vertical parabolas k)2 = Âą4c (x -
h)
Horizontal/Sideways parabolas
CONICS: Ellipses Standard equations for ellipses: (đ?‘Ľ − â„Ž)2 (đ?‘Ś − đ?‘˜)2 + =1 đ?‘Ž2 đ?‘?2 The a and b values are arbitrary, however, the a value is assigned to the larger denominator since this also indicates the major axis of the ellipse.
(đ?‘Ľ − â„Ž)2 (đ?‘Ś − đ?‘˜)2 + =1 đ?‘?2 đ?‘Ž2
(đ?‘Ľ − â„Ž)2 (đ?‘Ś − đ?‘˜)2 + =1 đ?‘Ž2 đ?‘?2
The equations for ellipses are often expressed in general form, and requires algebra to convert to the standard form.
CONICS: Ellipses See Pre-Test Problem: Find the major axis of an ellipse with equation 12x2 + 24y2 - 48x + 144y + 216 = 0. Standard equation of ellipse: (đ?‘Ľâ€“ â„Ž)2 (đ?‘Śâ€“ đ?‘˜)2 2 + = đ?‘&#x; đ?‘Ž2 đ?‘?2 Completing the square of given equation, đ?&#x;? 12x2 + 24y2 - 48x + 144y + 216 = 0 (đ?&#x;?đ?&#x;?) x2 + 2y2 - 4x + 12y + 18 = 0 [(x2 - 4x)+4] + [2(y2 + 6y)+9] = -18+4+18 (x2 – 4x + 4) + 2(y2 + 6y + 9) = 4 (đ?‘Ľâ€“ 2)2 (đ?‘Ś + 3)2 + = 1 4 2 Since a2 = 4 → a = 2 Major axis = 2a = 2(2) → 4
CONICS: hyperbolas Standard equations for hyperbolas: (đ?‘Ľ − â„Ž)2 (đ?‘Ś − đ?‘˜)2 − =1 đ?‘Ž2 đ?‘?2 The a and b values are arbitrary, however, the a value is assigned to the larger denominator since this also indicates the major axis of the hyperbola.
NOTES ON
trigonometry & geometry Like all engineering topics, the best way to understanding is by solving problems. The internet is full of sample problems that cater to your individual preference. Explore and be delighted in the responsibility!
Yours in academic empowerment, JPICHE-XU Department of Academics