DOUBLE-ANGLE AND HALF-ANGLE FORMULAS

Slides:

Advertisements

Similar presentations

Equations in Quadratic Form

Advertisements

Trigonometric Equations I

Operations on Functions

DOUBLE-ANGLE AND HALF-ANGLE FORMULAS. If we want to know a formula for we could use the sum formula. we can trade these places This is called the double.

Solving Quadratic Equations.

Parallel and Perpendicular Lines. Gradient-Intercept Form Useful for graphing since m is the gradient and b is the y- intercept Point-Gradient Form Use.

LINES. gradient The gradient or gradient of a line is a number that tells us how “steep” the line is and which direction it goes. If you move along the.

TRIGONOMETRY FUNCTIONS

If a > 0 the parabola opens up and the larger the a value the “narrower” the graph and the smaller the a value the “wider” the graph. If a < 0 the parabola.

PAR TIAL FRAC TION + DECOMPOSITION. Let’s add the two fractions below. We need a common denominator: In this section we are going to learn how to take.

Let's find the distance between two points. So the distance from (-6,4) to (1,4) is 7. If the.

REAL NUMBERS. {1, 2, 3, 4,... } If you were asked to count, the numbers you’d say are called counting numbers. These numbers can be expressed using set.

A n g l e s a n d T h e i r M e a s u r e. An angle is formed by joining the endpoints of two half-lines called rays. The side you measure from is called.

SETS A = {1, 3, 2, 5} n(A) = | A | = 4 Sets use “curly” brackets The number of elements in Set A is 4 Sets are denoted by Capital letters 3 is an element.

R I A N G L E. hypotenuse leg In a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle)

(r,  ). You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate.

SPECIAL USING TRIANGLES Computing the Values of Trig Functions of Acute Angles.

SOLVING LINEAR EQUATIONS. If we have a linear equation we can “manipulate” it to get it in this form. We just need to make sure that whatever we do preserves.

TRIGONOMETRIC IDENTITIES

You walk directly east from your house one block. How far from your house are you? 1 block You walk directly west from your house one block. How far from.

Logarithmic Functions. y = log a x if and only if x = a y The logarithmic function to the base a, where a > 0 and a  1 is defined: exponential form logarithmic.

INVERSE FUNCTIONS.

The definition of the product of two vectors is: 1 This is called the dot product. Notice the answer is just a number NOT a vector.

Dividing Polynomials.

exponential functions

GEOMETRIC SEQUENCES These are sequences where the ratio of successive terms of a sequence is always the same number. This number is called the common ratio.

VECTORS. A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude.

The standard form of the equation of a circle with its center at the origin is Notice that both the x and y terms are squared. Linear equations don’t.

ARITHMETIC SEQUENCES These are sequences where the difference between successive terms of a sequence is always the same number. This number is called the.

When trying to figure out the graphs of polar equations we can convert them to rectangular equations particularly if we recognize the graph in rectangular.

Properties of Logarithms

Logarithmic and Exponential Equations. Steps for Solving a Logarithmic Equation If the log is in more than one term, use log properties to condense Re-write.

The Complex Plane; DeMoivre's Theorem. Real Axis Imaginary Axis Remember a complex number has a real part and an imaginary part. These are used to plot.

Library of Functions You should be familiar with the shapes of these basic functions. We'll learn them in this section.

SEQUENCES A sequence is a function whose domain in the set of positive integers. So if I gave you a function but limited the domain to the set of positive.

11.3 Powers of Complex Numbers, DeMoivre's Theorem Objective To use De Moivre’s theorem to find powers of complex numbers.

COMPLEX Z R O S. Complex zeros or roots of a polynomial could result from one of two types of factors: Type 1 Type 2 Notice that with either type, the.

Sum and Difference Formulas. Often you will have the cosine of the sum or difference of two angles. We are going to use formulas for this to express in.

Solving Quadratics and Exact Values. Solving Quadratic Equations by Factoring Let's solve the equation First you need to get it in what we call "quadratic.

Surd or Radical Equations. To solve an equation with a surd First isolate the surd This means to get any terms not under the square root on the other.

COMPOSITION OF FUNCTIONS “SUBSTITUTING ONE FUNCTION INTO ANOTHER”

VECTORS. A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude.

DOUBLE- ANGLE AND HALF-ANGLE IDENTITIES. If we want to know a formula for we could use the sum formula. we can trade these places This is called the double.

Remainder and Factor Theorems. REMAINDER THEOREM Let f be a polynomial function. If f (x) is divided by x – c, then the remainder is f (c). Let’s look.

Dividing Polynomials Using Synthetic Division. List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put.

INTRODUCING PROBABILITY. This is denoted with an S and is a set whose elements are all the possibilities that can occur A probability model has two components:

The sum f + g This just says that to find the sum of two functions, add them together. You should simplify by finding like terms. Combine like terms &

Let's just run through the basics. x axis y axis origin Quadrant I where both x and y are positive Quadrant II where x is negative and y is positive Quadrant.

Solving Trigonometric Equations Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 x y π π 6 -7 π 6 π 6.

We’ve already graphed equations. We can graph functions in the same way. The thing to remember is that on the graph the f(x) or function value is the.

The sum f + g This just says that to find the sum of two functions, add them together. You should simplify by finding like terms. Combine like terms &

(r,  ). You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate.

TRIGONOMETRIC IDENTITIES

DOUBLE-ANGLE AND HALF-ANGLE FORMULAS

Systems of Inequalities.

RATIONAL FUNCTIONS II GRAPHING RATIONAL FUNCTIONS.

THE DOT PRODUCT.

INVERSE FUNCTIONS.

THE UNIT CIRCLE.

SIMPLE AND COMPOUND INTEREST

INVERSE FUNCTIONS Chapter 1.5 page 120.

THE UNIT CIRCLE.

INVERSE FUNCTIONS.

Symmetric about the y axis

exponential functions

Symmetric about the y axis

Presentation transcript:

DOUBLE-ANGLE AND HALF-ANGLE FORMULAS

If we want to know a formula for we could use the sum formula. we can trade these places This is called the double angle formula for sine since it tells you the sine of double 

Let's try the same thing for This is the double angle formula for cosine but by substiuting some identities we can express it in a couple other ways.

Double-angle Formula for Tangent

Summary of Double-Angle Formulas

We can also derive formulas for an angle divided by 2. Half-Angle Formulas As stated it is NOT both + and - but you must figure out where the terminal side of the angle is and put on the appropriate sign for that quadrant.

We could find sin 15° using the half angle formula. 30° 30° Since 15° is half of 30° we could use this formula if  = 30° 15° is in first quadrant and sine is positive there so we want the +

Let's draw a picture. 5 4   -3 Use triangle to find values.

If  is in quadrant II then half  would be in quadrant I where sine is positive 5 4   -3 Use triangle to find cosine value.

Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au