# Trigonometry 2: expressions and identities

**AS Level**

- Understand and use tanθ=sinθ/cosθ
- Understand and use sin^2θ+cos^2θ = 1
- Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle

**A Level**

- Sec
^{2}θ = 1 + tan^{2}θ and cosec^{2}θ = 1 + cot^{2}θ - Understand and use double angle formulae; use of formulae for sin(A±B), cos(A±B) and tan(A±B)
- Understand geometrical proofs of these formulae
- Understand and use expressions for acosθ + bsinθ in the equivalent forms of rcos(θ±α) or sin(θ±α)
- Construct proofs involving trigonometric functions and identities
- Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces

### Solving a Quadratic Equation in Sin(x)

These two Casio videos demonstrate how to solve the equation cos^{2}x +7sinx = 8 for x between 0 and 4π radians.

In the first video, the equation is rearranged to make the equation 6 cos^{2}x +7sinx - 8 = 0 then the identity sin^{2} x + cos^{2}x ≡1 is used to form a quadratic equation in one variable which is then solved.

The second video shows how a graphical calculator can be used to verify the solution. The first step is to use the calculator to draw the graph of y=6 cos^{2}x +7 sin x and the graph of y= 8. The the points of intersection of the two graphs are found, confirming the mathematical solutions.

### Using Trig Identities to Solve a Quadratic in Sin x

In the first of this pair of Casio videos, the identity sin^{2}x + cos^{2}x≡ 1 is used to solve fully the equation 9sin^{2}x + 21sinx = 3cos^{2}x -12.

The second video explores how a graphical calculator can be used to find solutions of the equation by plotting both sides of the equation we are trying to solve, and finding their points of interception.

### Trigonometry

This mathcentre collection includes resources that cover the **double angle formulae**, the **addition formulae**, **r cos theta **and more.

Each resource contains comprehensive notes, relevant examples and exercises for students wishing to independently consolidate their understanding of trigonometry.

### Trigonometrical Functions

This Further Thinking Questions resource, from Susan Wall, contains twelve problems requiring students to explore the properties of the functions of sine, cosine and tangent.

The problems are designed to encourage rich discussion in the classroom, by asking students to explain which is the odd one out, match functions with their graph, devise questions which give a certain set of solutions or state whether statements are true sometimes, always or never.

### Trigonometry 2

**RISP**s (**Ri**ch **S**tarting **P**oints) are intended to enrich the learning of mathematics in A Level classrooms.

This resource includes **Generating the Compound Angle Formula, **which requires students to generate a function by combining standard trigonometric functions and to graph the resultant function using graphing software. The activity leads to the compound angle formulae for sine and cosine.

### Using Rcos(x + α) to Find the Maximum and Minimum Values of a Function and to Solve a Trigonometric Equation

This resource contains three Casio videos that demonstrate how to use the form Rcos(x + α) to both find the maximum and minimum values of f(x) = 5sinx + 2cosx, and to solve f(x) = 1 within a given period.

The second video explains how to use the graphic calculator to find the coordinates of the maximum and minimum values of the given function. The function is drawn and the coordinates of the maximum and minimum points are verified.

The final video shows how to use graphic calculator to solve the equation 5sin x + 2cos x = 1 by drawing the graph of y = 5sin x + 2cos x and the graph of y = 1 and finding the points of intersection. The explanation includes instructions on how to set the axes to ensure the solutions are shown on the screen.

### Finding All the Solutions of Trigonometric Equations

These Casio videos explain how to solve the equation: sin(x+150) = 1/sqrt2 for x between 0⁰ and 360⁰ and how a graphical calculator can be used to verify the solution. The final video shows the mathematical solution to the equation cos(4x) = -1/2 and explains how the solutions found can be verified graphically by finding the points of intersection of the graphs y= cos(4x) and y= -1/2.

### aSin(x) + bcos(x)

This interactive excel file from The Virtual Textbook explores identities of the form aSin(x) ± bCos(x) and aCos(x) ± bSin(x) by plotting a series of interactive graphs. The resource works best when presented to students and adjusting the values of a and b to demonstrate the effect of the changes.

### Trigonometric Functions

The Spode Group produced this A-level mathematics textbook to show the practical importance of trigonometric functions. The booklet contains five case studies, including c**ylindrical oil container **that covers the applications of sine, cosine and tangent and **pull on the arms of a windsurfer** which combines kinematics and trigonometry when resolving forces in equilibrium.

The resource provides a variety of contexts for mathematical modelling and contains a variety of 'real-world' problems.