Chapter+4+Notes

Ryan Sullivan Precalc February 12, 2012

Similar Triangles- similar in math means identical in shape, but not necessarily the same size. Similar triangles have the same exact angle measures and shape but different side lengths

Right Triangle- a triangle in which one of the angles is 90 degrees, a right angle. the longest side of the triangle is the HYPOTENUSE which is opposite the right angle, the other two sides are referred to as the LEGS.

Pythagorean Theorem relates to the sides of a right triangle. says the sum of the squares of the lengths of two legs is equal to the square of the length of the hypotenuse:

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**__Right Triangle Ratios__**

If we know the lengths of a side of one triangle we know the lengths of certain sides of a similar triangle

In similar triangles, the sides opposite corresponding angles must be proportional so the following ratios hold true:

a/a' = b/b' = c/c'

Angle Measurements: 1. degree 2. radian

to convert: degree x 3.14/180 = pi radian

Coterminal Angle: share the same terminal, or point.



sin30= 1/2 sin150= 1/2 - they are the same because they are coterminal angles

Angular velocity and linear speed....

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Rationalizing Any Denominators containing a Radical:

sin0= 5/sq29 x sq29/sq29 = 5sq29/29

Algebraic Signs of Trigonometric Functions:

The x-coordinate in positive quadrants I and IV and negative quadrants II III The y-coordinate is positive in quadrants I and II and negative in quadrants III and IV

Quadrant Positive Trig Function I all three

II Sine

III Tangent

Cosine Cosine

Working with values of Trig functions for quadrantal functions:

Ex. cos540 + sin270

The terminal side of an angle with measure 540 540-360=180 Lies along the negative x-axis

Evaluate Cosine of an angle whose terminal side cos540= -1 lies along the negative x-axis

Evaluate sine of an angle whose terminal side sin270= -1 lies along the negative y-axis

Add the two values -1+-1= -2



4.4

The Law of Sines:

For a triangle with sies lengths a, b, and c, and opposite angles of measures, A, B, y the following is true

__sinA__ = __sinB__ = __sinY__ a b c



In other words, the ratio of the sine of an angle in a triangle to its opposite side is equal to the ratios of the sines of the other two angles to their opposite sides

Four ways to solve triangle

AAS or ASA= angle, angle, side SSA= side, side, angle SAS= side, angle, side SSS= side, side, side

We can solve any triangle given three measures as long as one of the measures is a side length. Depending on the info given, we either apply the law of sines:

sinA = sinB = sinC a b c

an the angle sum identity, or we apply a combo of law of cosines:

a^2= b^2 + c^2 - 2bcosA

- to find for the other values replace a, b, and c values in the different parts of the equation

This table summarizes how to solve:

Step 2. find the remaining sides with Law of Sines || Step 1. apply law of sines to find first angle Step 2. find the remaining angle with A+B+C=180 Step 3. find the remaining angle with law of sines || Step 2. find either remaining angles with law of sines Step 3. find the final angle with A+B+C=180 ||
 * AAS or ASA || two angles and a side || Step1. find the remaining angle with A+B+C= 180
 * SSA || two sides and an angle opposite of one of the sides || This is and ambiguous case, so there is either no triangle, one triangle, or two triangles. If the given angle is obtuse then there is either one or no triangle. if the given angle is acut, then there is no triangle, one triangle, or two triangles
 * SSS || three sides || Step 1. find the largest angle with law of cosines