Trigonometric+Functions+and+their+Derivatives-2

History [|Trigonometry] was first compiled by the Greek mathematician [|Hipparchus] in the 2nd century BCE. The early study of triangle can be traced to Egyptian and Babylonian mathematics in the 2nd millennium BC. The study of trigonometric functions began with [|Hellenistic mathematics], and was first associated with Hellenistic Astronomy, especially in India. The study of trigonometric functions began during the Gupta period in part due to Indian astronomy. Trigonometric functions continued to be studied in Islamic mathematics until it was became a spate subject beginning with the Renaissance. Modern trigonometry began in the Age of Enlightenment when [|Isaac Newton]and [|James Stirling], and was continued by [|Leonhard Euler.]
 * Trigonometric Functions and their Derivatives ||

In mathematics trigonometric functions are functions of an angle. They relate the length of the sides of a triangle to the angles. The most common trigonometric functions are sine, cosine and tangent, though cosecant, secant, and cotangent are also commonly used. The standard unit circle has a radius of 1 and in that context the sine of an angle gives the y-value(rise) of the created triangle and cos gives the x-value(run) of the created triangle. media type="youtube" key="6Qv_bPlQS8E" height="279" width="339" align="right"
 * sine ( q ) = opp/hyp || [|cosecant]( q ) = hyp/opp ||
 * cosine ( q ) = adj/hyp || [|secant]( q ) = hyp/adj ||
 * [|tangent]( <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">q <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">) = opp/adj || <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">[|cotangent]( <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">q <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">) = adj/opp ||

The most important trigonometric functions are sin and cos, all others can be derived from those two. tan( q ) = sin( q )/cos( q ) csc( q ) = 1/sin( q ) sec( q ) = 1/cos( q ) cot( q ) = 1/tan( q )

**Reciprocal identities**

**Pythagorean Identities** **Quotient Identities** **Co-Function Identities** **Even-Odd Identities**

**Sum-Difference Formulas**

**Double Angle Formulas** **Sum-to-Product Formulas** **Product-to-Sum Formulas**
 * Power-Reducing/Half Angle Formulas **

<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 10pt;">A[| cofunction]of a given trig function (f) is, the function obtained after the complement its parameter is taken. Since the complement of any angle <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 10pt;">q <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 10pt;"> is 90° - <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 10pt;">q <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 10pt;">, then <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt; text-indent: 0.5in;">sine(90° - <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 0pt; text-indent: 0.5in;">q <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt; text-indent: 0.5in;">) = cosine( <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 0pt; text-indent: 0.5in;">q <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt; text-indent: 0.5in;">)

<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt; text-indent: 0.5in;">secant(90° - <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 0pt; text-indent: 0.5in;">q <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt; text-indent: 0.5in;">) = cosecant( <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 0pt; text-indent: 0.5in;">q <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt; text-indent: 0.5in;">)

<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 6pt; text-indent: 0.5in;">tangent(90° - <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 6pt; text-indent: 0.5in;">q <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 6pt; text-indent: 0.5in;">) = cotangent( <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 6pt; text-indent: 0.5in;">q <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 6pt; text-indent: 0.5in;">)

<span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">= <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">q || <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">[|arccosecant](hyp/opp) <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">= <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">q || <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">= <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">q || <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">[|arcsecant](hyp/adj) <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">= <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">q || <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">= <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">q || <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">[|arccotangent](adj/opp) <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">= <span style="color: black; font-family: Symbol; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">q ||
 * <span style="color: black; display: block; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt; text-align: center;">[|inverse functions] ||
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">[|arcsine](opp/hyp)
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">[|arccosine](adj/hyp)
 * <span style="color: black; font-family: 'Times New Roman','serif'; font-size: 16px; line-height: normal; margin: 0in 0in 0pt;">[|arctangent](opp/adj)

Below are the graphs of the sin, cos and tan graphs.

media type="youtube" key="OLzXqIqZZz0" height="218" width="239" align="right"

// Proof that the derivative of sin x is cos x: //

// f // (x) = sin x

// f // ′(x) = lim __sin (x **+** h) **–** sin x__ h ® 0 h

sin A **–** sin B = 2 cos ½ (A **+** B) sin ½ (A **–** B)

// f // ′(x) = lim __2 cos ½ (x **+** h **+** x) sin ½ (x **+** h **–** x)__ h ® 0 h

// f // ′(x) = lim __2 cos (x **+** h/2) sin (h/2)__ h ® 0 h

The limit of a product is the product of the limits: // f // ′(x) = lim cos (x **+** h/2) ∙ lim __2sin (h/2)__ h ® 0 h ® 0 h  // f // ′(x) = cos x ∙ lim __sin (h/2)__ h ® 0 h/2 Since lim __sin x__ = 1, //f// ′(x) = cos x ∙ 1 h ® 0 x

// f // ′(x) = cos x  // Proof that the derivative of cos x is //// - //// sin x: //

// g // (x) = cos x  // g // (x) = sin ( p /2 ** - ** x)

// g // ′(x) = cos ( p /2 ** - ** x) ∙ ( ** - ** 1) // g // ′(x) = ** - ** cos ( p /2 ** - ** x)  // g // ′(x) = ** - ** sin x[|tangent]

Works Cited

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