10 2 sum and diff formulas for tangent
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1) Formulas are derived for tangent of the sum and difference of two angles using previous trigonometric identities: tan(α + β) = (tanα + tanβ) / (1 - tanαtanβ) and tan(α - β) = (tanα - tanβ) / (1 + tanαtanβ). 2) These formulas are valid when the tangents of the individual angles and their sum/difference are defined, which excludes cases where the cosine of any angle is 0. 3) Examples are provided to demonstrate applying these formulas to find the exact value of tangents of sums/differences of known angles.
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4.4 multi step trig problemshisema01 This document provides instructions for solving for the length of a shared side (x) of two triangles. It instructs the user to write two equations, one for each triangle, relating the lengths of the sides including x. It also describes an approach for solving when the triangles form one large non-right triangle, which is to draw the altitude splitting it into two triangles and using the fact that triangle angles sum to 180 degrees. The user is then told to solve the two equations to find the value of x.
6 5 parabolas
6 5 parabolashisema01 This document provides an overview of parabolas including their key properties and examples of how to graph and write equations of parabolas. It defines that every point on a parabola is equidistant from the focus and directrix, with the directrix perpendicular to the line of symmetry and the vertex halfway between them. Examples are given of finding the focus, directrix, and graphing parabolas given these features as well as writing the equation of a parabola given its directrix.
4.3 finding missing angles
4.3 finding missing angleshisema01 Inverse trig ratios, including sin-1, cos-1, and tan-1, are used to solve for missing angle measures. When given a decimal value for sin, cos, or tan, the corresponding inverse trig function is used to find the angle measure to the nearest degree. When given a triangle, the appropriate trig ratio is set up and the inverse function is then used to find the measure of the indicated angle to the nearest degree.
4.2 solving for missing sides
4.2 solving for missing sideshisema01 This document provides instructions for solving for a missing side of a triangle using trigonometric functions. It tells the reader to label the sides of the triangle, set up an equation using the information given, and solve for the variable to find the missing side, making sure a calculator is in degree mode.
4.1 trig ratios
4.1 trig ratioshisema01 This document defines and explains the three main trigonometric ratios - sine, cosine, and tangent - which are used to solve for missing angles and sides of right triangles. It identifies the hypotenuse as the longest side, and the adjacent and opposite sides in relation to a given angle. The document provides the ratio equations and examples of calculating trig ratios for various angles in a right triangle.
R.4 solving literal equations
R.4 solving literal equationshisema01 This document defines a literal equation as an equation with two or more different variables. It provides examples of literal equations such as A = bh, where b is solved for, and P = 2L + 2W, where W is solved for. The document demonstrates how to solve various literal equations for the indicated variable, such as solving for P in I = Prt, solving for x in A = x + y^2, and solving for x and y in two additional equations.
R.3 solving 1 var inequalities
R.3 solving 1 var inequalitieshisema01 This document provides instructions for solving inequalities and graphing their solutions. It notes that when multiplying or dividing an inequality by a negative number, the inequality symbol is reversed. Open circles are used for < and >, while closed circles are used for ≤ and ≥. Examples are provided of solving various inequalities algebraically and graphing the resulting solution sets on a number line.
R.2 solving multi step equations
R.2 solving multi step equationshisema01 Multi-step equations involve more than two steps to solve. To solve them, simplify the side with variables by distributing and combining like terms, then solve as a single-step or two-step equation. The key steps are to distribute any parentheses, collect like variable terms on one side and constants on the other, and use opposite operations to solve for the variable.
R.1 simplifying expressions
R.1 simplifying expressionshisema01 Algebraic expressions contain numbers, variables, and operators and can be simplified using the distributive property and by combining like terms. The document provides examples of algebraic expressions that can be simplified by first distributing all terms inside parentheses and then combining like terms, such as 3(4n + 2) + n, which is simplified to 12n + 6 + n = 13n + 6.
10 4 solving trig equations
10 4 solving trig equationshisema01 This document discusses two methods for solving trigonometric equations: graphing and algebraically. Graphing allows for visually seeing solutions but may result in decimal answers, while algebraic solving provides exact solutions but can become messy. It provides tips for algebraic solving based on whether the equation contains functions of 2x and x, or only 2x. Examples are given to demonstrate each method.
7 2 adding and subtracting polynomials
7 2 adding and subtracting polynomialshisema01 The document discusses several methods for adding and subtracting polynomials: using algebra tiles, the horizontal method, and the vertical method. It provides examples of adding and subtracting polynomials with one and two variables. Terms with the same variables are combined by adding the coefficients. For subtraction, the signs of the terms in the subtracted polynomial are changed before adding.
7 3 multiplying polynomials
7 3 multiplying polynomialshisema01 This document discusses four methods for multiplying binomial expressions: algebra tiles, the distributive property, a table or "box" method, and FOIL. It explains that every term in the first binomial must be multiplied by every term in the second binomial. Examples are provided of multiplying binomials such as (x + 2)(x + 5), along with practice problems like (x + 3)(x - 4) and (2p - 4)(3p + 2).
7.2 simplifying radicals
7.2 simplifying radicalshisema01 The document discusses simplifying radical expressions. It defines key terms like the radical sign and radicand. It provides two methods for simplifying radicals: using the product property to rewrite with the largest perfect square factor or making a factor tree to pull out factors. Examples are provided to demonstrate simplifying radicals of various forms, including those with variables and higher root expressions. The document also contains practice problems for simplifying radical expressions.
10 3 double and half-angle formulas
10 3 double and half-angle formulashisema01 1) This document discusses double-angle and half-angle formulas for trigonometric functions like sine, cosine, and tangent.
2) It derives formulas that relate trig functions of double and half angles to trig functions of the original angle, such as sin(2x) = 2sin(x)cos(x) and sin(x/2) = ±√(1-cos(x))/2.
3) Examples are provided to demonstrate applying these formulas to simplify trigonometric expressions and derive new identities.
6.7 other methods for solving
6.7 other methods for solvinghisema01 This document discusses two additional methods for solving quadratic equations: the quadratic formula and graphing. The quadratic formula can be used to solve any quadratic equation in standard form (ax^2 + bx + c = 0) by inputting the a, b, and c coefficients. Graphing involves changing the equation to y= form, plotting it on a calculator to find the x-intercepts, which are the solutions. Both methods are demonstrated with examples.
6.7 other methods for solving
6.7 other methods for solvinghisema01 This document discusses two additional methods for solving quadratic equations: the quadratic formula and graphing. The quadratic formula can be used to solve any quadratic equation in standard form (ax^2 + bx + c = 0) by inputting the a, b, and c coefficients. Graphing involves changing the equation to y= form, plotting it on a calculator to find the x-intercepts, which are the solutions. Both methods are demonstrated with examples.
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- 2. Deriving tan(α + β) Using previous formulas:
- 4. Deriving tan(α - β) We know that tan(-β) = - tan β So, replacing β with – β gives:
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