Trig Toolkit. 30 = π/6. 0 or 360 =2π. 330 = 11π/ = 7π/4. Note: In the table below, r can be replaced with. + y, since. P θ θ.

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1 Trigonometr Toolkit 90 / 0 / 5 /4 50 5/ / 45 /4 0 /6 0 or Triangles: or an multiples of Triangles or an multiples of. 0 7/6 5 5/4 40 4/ 70 / 00 5/ 0 /6 5 7/4 Conversions: degrees radians radians degrees Acute angle α is the reference angle for a (80 α) angle QII, a (80 + α) angle QIII, a (60 α) angle QIV as well as all of the coterminal angles, (α ±60n) P(,) r Note: In the table below, r can be replaced with r + +, ce ( cos, ) P θ θ P(cos 0, 0) basic other reciprocals Sine (opp/hp) Cosecant θ r, r 0 csc θ r cscθ, 0 θ Coe (adj/hp) cos θ r, r 0 Tangent (opp/adj) tan θ, 0 Secant sec θ r secθ, 0 Cotangent cot θ cotθ, 0 tanθ r r Rad. cos θ θ tan θ Undef Undef. 0 0 Now just use the reference angles! Positive Values? Sin All ( 0, ) /, /, Tan Cos /4 /6, 0 (,0 ) S. Stirling Page of 8

2 Relationships Among the Functions Reciprocal Relationships: cscθ θ secθ cotθ tanθ Relationships with Negatives: csc tan ( θ ) θ and cos ( θ ) ( θ ) cscθ and sec( θ ) ( θ ) tanθ and cot ( θ ) Quotient Identities: θ tanθ, 0 cotθ, θ 0 θ secθ cotθ Pthagorean Relationships: θ + cos θ + tan θ sec θ + cot θ csc θ Double-Angle Formulas (These will be given to ou b AP; however, our book won t do this.) ( α) α cos α cos( α) cos α α cos( α) α cos( α) cos α tanα tan( α) tan α Periodicit Relationships: ( θ + ) θ (same for csc.) cos( θ + ) (same for sec.) tan( θ + ) tanθ (same for cot.) Definition Periodic Function: f() is periodic if there is a positive number p such that f( + p) f() for ever value of. The smallest such value of p is the fundamental period of f. S. Stirling Page of 8

3 Graphs of the Trigonometric Functions For all functions, n an integer. Sine θ. Period: Coe. Period: Domain: R / / / / Domain: R / / / / Range: f ( ) f ( ) Range: f ( ) f ( ) Cosecant cscθ. θ Secant secθ. Period: / / / / Period: / / / / Domain: 0 + n Range: f ( ) f ( ) or f ( ) Domain: + n Range: f ( ) f ( ) or f ( ) Tangent θ tanθ. Period: Domain: + n Range: R / / / / Cotangent cotθ tanθ cotθ θ Period: Domain: 0 + n Range: R / / / / Even and Odd Functions: cos() and sec() are even functions. (), csc(), tan() and cot() are odd functions. S. Stirling Page of 8

4 Sine and Coe Functions and Modeling Domain: R Range: R, where -a f() a Period: How to write models: a cos [b( c)] + d Determine the amplitude, a, vertical stretch or compression: a (maimum minimum) Distance from midline to the maimum or minimum. Determine the horizontal stretch or compression, b: b / period. Period the horizontal distance between successive maimums (or minimums). Determine the horizontal shift, c: Use the -value of a maimum value (or minimum value) and figure the number of -units needed to map the parent function onto the transformed function. Determine the vertical shift, d, location of the midline: d average of the maimum and minimum value, midpoint. The midline is also called the ais of the wave. Graph transformations Eample: cos(4 ) 7 Must get it in the correct form! a cos [b( c)] + d cos 4 7 Stretch & Compress cos( ) cos( ) vertical stretch (factor of ) or amplitude cos(4 ) horizontal compress (factor of 4) or period 4 Stretch factor 4 Compress factor 4 Translations cos(4 ) 7 translation shift down 7 or midline at 7 cos 4 7 translation shift right or down 7 4 / / / / / / / / right / 0 This process works for an transformations on an function. S. Stirling Page 4 of 8

5 Graphs of the Inverse Trigonometric Functions With inverses, the roles of the and are interchanged. You must limit the domains of the original (to get a function) and then reflect over the line in order to get the inverse function. Note: the ranges of the inverse functions are the limited domain of the original! Inverse Trigonometric Functions graph splits the -ais (works around 0) ( ) or arc ( ) cos ( ) / graph above the -ais (works around /) or arccos ( ) / Domain -values IN value, v (are ranges of trig. functions) csc ( ) ( ) 0 / / / sec ( ) cos ( ) / or or tan ( ) ± / ( ) tan ( ) cot 0 or Range -values OUT angle θ (Limited domains of trig. functions) / θ in Quadrant I or IV / 0 θ in Quadrant I or II The graph reflects over the -ais. Helpful website: Lesson 0 is good for Trig Functions! S. Stirling Page 5 of 8

6 Moving From Sine to Arce v θ Concept of inverse: ( ) so ( v) θ or arc ( v) θ, think the angle whose e is value Inverse Sine: Graphicall To make the reflection over a function, an inverse, ou must limit the domain of e. (Which limits the range of arce.) ( ) D: (, ) R: [,] Negative θ Q III Q IV Q I Positive θ Q II / / Limit the domain, the angle values θ. Interchange and & solve for. ( ) ( ) ( ) D: [,] Pos. θ Neg. θ / R:, / / Q I Q IV Inverse Sine: Numericall Interchanging the and values. ( ) 0 0 ( 0) ( ) is out of the domain. 6 4? ( ) is out of the range. Inverse Sine: Unit Circle Arc onl in Quadrant I and Quadrant IV because of the limited domain. θ ask ( ) Sine positive in Quad I, so θ ask ( ) θ and θ Sine negative in Quad IV, so 4 and 4 4,, S. Stirling Page 6 of 8

7 Inverse Trigonometric Functions (continued) Inverses: ( θ ) value so ( ) ( ) Composites: ( ) value θ when value value as long as value θ and value. and ( ( θ) ) θ as long as θ Limit domain of e to get a function. The undoing works so long as the domain and range of the inverse trig functions are met (see page 5). So when simplifing trig. epressions or solving trig. equations, ou must be aware of the restrictions on the input variable. E. Evaluate, the angle whose sign is ( θ ) Onl in Q IV, so. 6 It works the same wa for the other inverse trig functions. See the inverse trig. chart. implies cos 0 when 0. So, cos 0 E. cos ( 0) and. E. tan implies tan when. So, tan. The easiest wa to do E 4 E 6 is to draw right triangles. Keep in mind what quadrant ou are in (determined b the range of the inverse trig function). E 4. Evaluate cot θ Quad. IV, e negative. Set up triangle ug arc: E 5. Solve in ( ) ( ) s cos both positive in Quad. I, so take e both sides cos ( ) and set up the triangle ug arccos: Use Pthagorean thm. + b b Cotangent is adj/opp cot [ θ ] OR ( ) 6 c ot 6 Sine is opp/hp, so θ now substitute, 0 watch the domains: Now solve for. radicals 0, 0 inverse coe between and., S. Stirling Page 7 of 8

8 E 6. Write sec ( ) in algebraic form. ( ) sec in Quad. I or II because of restrictions on secant 0 θ, θ sec θ, Set up triangle, secant is hp/adj. Use Pthagorean thm. + b b Sine is opp/hp and e would be positive So θ, the guarantees θ positive. To help ou remember our relationships. Cosecant is defined as csc. The are reciprocals. (The inverses are not!) ( ) Wh is csc? Start with csc, where and, 0 So csc and we know cscθ, replace csc θ now solve for and now undo e and csc Which means csc. This is NOT the same as csc!!! Careful! S. Stirling Page 8 of 8