MAT170 LIST OF CONCEPTS TEST 3
3.1 – Introduction:
· radian measure.
3.2 The sine and cosine functions
· Definition of sin t and cos t on the unit circle
· Pythagorean identity
· Periodic functions: understand the concept of a function that repeats its output on a regular basis (the period).
· Know the EXACT values of sine and cosine of p/3, p/4 and p/6 and how to derive values for integer multiples of these angles (e.g. 5p/3, 11p/6, -7p/4 …)
· Reference number
· Trigonometric equations
3.3 Graphs of sine and cosine functions
· Graphs of sine and cosine: cosine function is even, the sine function is odd. Sine and cosine are periodic of period 2p.
· Sinusoidal functions (see box on page 208): know the meaning of A, B and C: |A|= amplitude, 2p/|B| = period, |C/B|=horizontal shift.
· Know how to sketch one period of the graph given the formula and how to find a possible formula given the graph.
3.4 Other trigonometric functions
· Know the definitions of tangent, cotangent, secant and cosecant functions.
· Know the graph of the tangent function (tangent is periodic of period p since it is the slope of the ray from the origin of an angle t).
3.5 Trigonometric identities
· Important identities: Pythagorean identities (page 225), Sum and difference formulas for the sine and cosine (pages 223/224), Double angle formulas (page 228), Half angle formulas (page 230), Formulas for products of sine and cosine (page 232).
· Know how to use sum/difference formulas and double angle formulas for sine and cosine to find exact values of trig functions of special angles and to prove simple identities relating sine and cosine.
· Know how to solve trig equations using identities.
· Know how to verify (prove) an identity algebraically and graphically.
Sum/difference formulas, half- angle formulas and formulas for products of sine and cosine will be given on the test. Pythagorean identities and double angle formulas will not be given. You need to memorize them (or be able to derive them). Note that double angle formulas can be easily derived from sum/difference formulas.
Strategies for proving an identity:
1. Work separately on the 2 sides of the identity. Begin with the more complicated expression and modify it using algebra and known identities so that it looks like the other side.
2. If no other move suggests itself, convert the entire expression to one involving sine and cosine.
3. Combine fractions by writing them over a common denominator.
4. Use the algebraic identity to set up applications of the Pythagorean identities (e.g. )
5. Always be mindful of the “target” expression and favor manipulation that brings you closer to your goal.
3.6 Right angle trigonometry
· Trigonometric functions of an angle in a right triangle.
· Measuring angles using degrees
· Conversion between radians and degrees
3.7 Inverse trigonometric functions
· Know the definitions of inverse sine, cosine and tangent functions. Think of as the angle in [] whose sine is x. Think of as the angle in [0,p] whose cosine is x. Think of as the angle in [] whose tangent is x.
· KNOW YOUR BASIC VALUES FOR SINE, COSINE AND TANGENT IN EXACT FORM! E.g. radians (Examples 4,5,6).
· Know the domain and range of the inverse trigonometric functions.
· Arcsine and arctangent are odd functions, arccosine is neither even nor odd.
· Know how to solve a trigonometric equation graphically.
· Know how to use the inverse sine, cosine and tangent functions to solve trigonometric equations. Remember, these functions will only give one value back if you use your calculator, so you will need to use your knowledge of the sine, cosine and tangent graphs to determine the other solutions, if they are applicable.
3.8 Applications of trigonometric functions
· Know the Law of Cosines and the Law of Sines (these formulas will not be given in the test) and how to apply them.
· Sine combination formulas, cosine combination formulas (optional)
· Herone’s formula (optional)