Ella's Blog

5. Trigonometry

-Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.

-Sine, cosine and tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle.

The functions of sin, cos and tan can be calculated as follows:

Sine Function: sin(θ) = Opposite / Hypotenuse.

Cosine Function: cos(θ) = Adjacent / Hypotenuse.

Tangent Function: tan(θ) = Opposite / Adjacent

-We can find the coterminal angles of a given angle by using the following formula

:Coterminal of θ = θ + 360° × k

–The reference angle is the positive acute angle that can represent an angle of any measure.

:reference angle = 180° – angle , 180° to 270°: reference angle = angle – 180° , 270° to 360°: reference angle = 360° – angle .

–The Law of Sines (or Sine Rule) is very useful for solving triangles:

a /sin A = b/ sin B = c/ sin C

-<The law of cosines>

a, b and c are sides. C is the angle opposite side c. The Law of Cosines (also called the Cosine Rule) says: c² = a² + b² − 2ab cos(C)

5.1 Angles in standard position in quadrant 1

• r in term of x and y
• the value ofθin terms of x and y
•  the x-coordinate of P in terms of r andθ
• the y-coordinate of P in terms of r and θ

The coordinates of a point P on the coordinate plane can be described by its distance r from the origin, O, and the angleθthat Op makes with the positive x-axis. When the angle θ, between 0and 360 is measured counter clockwise from the positive x-axis, the angle is in standard position. The ray OP is the terminal arm of the angle and the oint P is a terminal point for the angle.

Trigonometry is essential to navigation. A direction can be described by relating it to two of the compass points: north, south, west and east.

ex) a heading of W30S means from a direction due west, rotate 30 counter clockwise; that is, toward south.

ex)a heading of W40N means from a direction due west, rotate40 clockwise; that is, toward north.

5.2 Angles in standard position in all quadrant

The terminal arm of an angle in quadrant 1 can be successively reflected in both axes to form 4 different angles in standard position. The reference angles in standard position. The reference angle for all angles is the acute angle that the terminal arm makes with the x-axis.

In Lesson 5.1, the trigonometric ratios of an angle in standard position in Quadrant 1 were related to the coordinates of a point on the terminal arm of the angle. These relationships can be extended to define the primary trigonometric ratios for any angle θ  in standard position. For angles greater than 90, a represents the reference angle.

Lesson 6.5 – solving rational equations

To solve an equation with rational coefficients, the fractions can be cleared by multiplying both sides of the equation by a common denominator.

The same strategy can be used to solve an equation that contains rational expressions, known as a rational equation.

First identify the non-permissible values of the variable, then when the equation has been solved, check to see that the solution is a permissible value/ If it is not, the solution is an extraneous root.

All solutions of equations should be verified by substituting in the original equation. These verifications may not be included in the material that follows in this text.

Lesson 6.6 – applications of rational equations

Rational equation can be used to solve variety of real-world problems, including those involving motion, work, and proportions.

Lesson 6.1 Equivalent Rational Expressions

When the numerator and denominator of a fraction are integers, the fraction is a rational number.

When the numerator and denominator of a fraction are polynomials, the fraction is a rational expression.

• A rational expression is an algebraic expression that can be written as the quotient of two polynomials

Rational expressions are not defined for values of the variable that make the denominator 0. These values are called non-permissible values.

Lesson 6.2 – Multiplying and dividing rational expressions

The strategies for multiplying and dividing rational numbers can be used to multiply and divide rational expressions. All non-permissible values of each expression being multiplied or divided must be stated.

Lesson 6.3 – Adding RE with monomial denominators

The strategies for adding and subtracting rational expressions: written the expressions with a common denominator, then add or subtract the numerators. Identify non=-permissible values of the variables

Lesson 6.4 – Adding RE with polynomial denominators

The strategies for adding and subtracting rational expressions with monomial denominators can be used to add and subtract rational expressions with binomial and trinomial denominator.

Lesson 4.8 – solving inequalities using graphing

A single-variable linear inequality is an inequality where one side of the inequality is a linear expression and the other side is either a constant or another linear expression;

ex) A linear inequality has one of the  following formats when written in general form.

mx +b> 0

mx+b≥ 0

mx+b < 0

mx +b≤0

For a linear inequality to exist, m≠0

The solution of each inequality is an interval, a continuous set of x-values that make the inequality true.

Linear inequalities can be solved by graphing.

ex) to solve the linear inequality 3x-1>2

: Graph y=3x-1 and y=2 on the same grid.

The graph of y=3x-1 has slope 3 and y-intercept -1.

The graph of y==2 is a horizontal line with y-intercept 2.

• A single-variable quadratic inequality has a quadratic expression on one side of the inequality and a constant, a linear expression, or another quadratic expression on the other side of the inequality.

Lesson 4.9 Solving Linear and quadratic Inequalities Algebraically

• The intervals in Construct Understanding are written using inequality notation. Another way to describe intervals is with interval notation. The table below illustrates intervals using number lines, inequality notation, and interval notation. a and b are constants and they are the critical values.
• An interval written as (a,b) is an open interval.

An interval written as [a,b] is a closed interval

An interval written as [a,b) and (a,b] is a half-open (or a half closed) interval

<Midterm Review>

Ch.1 Roots and Powers

• Natural Numbers
• Whole Numbers
• Integers
• Rational Numbers
• Irrational Numbers

n is the index

a is the radicand

n√ is the root symbol

n√a is the radical

Ex)

Ch.2 Radical Operations and Equations

Recall the multiplication property of radicals

n√ab =n√a x n√b, where n ∈ N and a,b, n√a, n√b ∈ R

This property is used to  write radicals in different forms. In the same way that 3 is the coefficient of the algebric term, 3x, we say that 3 is the coefficient of the mixed radical, 3√7.

Ch.3 Solving Quadratic Equations

Factoring

1 – one thing in common GCF a (b+c)

2 – two terms -> difference of squares (a+b)(a+b)

3- three terms -> pattern x^ x # product and sum

ex)

Ch.4 Analyzing quadratic Functions and Inequalities

1. units left –
2. Units right –
3. units up –
4. units down –

standard Form

vertex = (p,q)

axis of symmetry = x=p

ex) = minimum value

= maximum value

Congruent =

*standard Form-

general Form –

Factored Form –

4. Analyzing Quadratic Functions and Inequalities

• The effect of changing q in y=x^+q
• The effect of changing p in y=(x-p)^2
• The effect of changing a in ax^2

When these three transformations are combined, the resulting equation is the standard form of the equation of a quadratic function.

y=a(x-p)^2+q

The coordinates of the vertex are (p,q)

ex)

y=2x^2+ 16x + 24

1. 2(x^2+8x) +24
2. y=2(x^2 +8x+16-16) +24
3. y=2(x^2 +8x +16) -2(16) +24
4. y=2(x^2 +8x+16)-32 +24
5. y=2(x+4)^2 -8

vertex (-4,-8)

4. Analyzing Quadratic Functions and Inequalities

y=x^2

y=-x^2

y=x^2+2x+1

y=-x^2+2x

• The vertex of a parabola is its highest of lowest point. The vertex may be a minimum point or a maximum point.
• The axis of symmetry intersects the parabola at the vertex. The parabola is symmetrial about this line.
• The x-intercepts of the graph of a quadratic function, y=ax^2 +bx+c, are called the zero of the fuction

y=a(x-p)^2+q

a=opens up/down, stretch IaI>1, compression IaI<1

p=horizontal translation (vertex) left<–> right

q=vertical translation (vertex) up <–> down

Factoring

1-one thing in common GCF a (b+c)

2-two terms – different of squares (a+b)(a+b)

3-three terms- pattern x^ x # product and sum

3.5 Developing and Applying the Quadratic Formula

In 3.5 I learned how to solving a Quadratic Equation of the Form x^2+ bx+c=0

x=(-b±√(b^2-4ac))/2a

This equation called the quadratic formula and it can be used to determine the solution of any quadratic equation written in the form axsqure+bx+c=0

3.6 Interpreting the Discriminent

*Number of Roots of a Quadratic Equation

The quadratic equation axsquare+bx+c=0

• two real roots when b^2 -4ac>0
• exactly one real root when b^2 -4ac =0
• no real roots when b^2 -4ac<0

Factoring

1 – one thing in common GCF a (b+c)

2 – two terms -> difference of squares (a+b)(a+b)

3- three terms -> pattern xsqure x # product and sum

3.1 Factoring trinomials 

In 3.1, I learned how to factor trinomials by finding the factors of X^2 and the coefficient.

In step 1, find factors of 5 and -8 separately, then arrange in these numbers in two brackets, so then when you factor out the it, you would end up with the same trinomial.

In step 2, make sure that the product of 8 and -1 equals to -8.

In step 3, make sure that the product of 5x and x equals to the first firm of the trinomial.

In step 4, make sure to multiply the two sets of two numbers separately and add them together to check if this result equals to the second term of the trinomial.

3.2 Factoring Polynomial Expressions

In this section, I learned how to factor by perfect squares.

For instance, I have to find the root of each term. In step 1, square root each term.

In step 2, I would place the roots of the first term in the binomial in the first place in each bracket. (ex. x/10)

In step 3, I placed the roots of the second term in the binomial in the second place in each bracket. However, one term must be positive and the other must be negative.

3.3 Solving quadratic equations by factoring.

In this section, I learned how to solve quadratic equations by factoring and verify the solutions.

In this example, first factor the trinomial (step 1)

Then, substitute 1 into X and verify if the equation equals to 0 as well.

In step 3, very if the equation equals to 0 as well by substituting X=5 in to X.

3.4 Using square roots to solve quadratic equations. 

In this section, I learned how to solve for the unknown constant in quadratic equations.

In this example, first divide the coefficient of the second term in the binomial by 2.

In step 2, place this (2.5) in to a bracket as a constant and square the bracket.

In step 3, solve for the constant of this equation by multiplying 2.5 by 2.5. (ex. 6.25)