Math Trigonometry Review CHSN Review Project This review guide was written by Dara Adib. It was designed for a test on trigonometry administered by Jeanine Lennon to her Math 12H (4H/Precalculus) class on February 12, 2008, but also applies to trigonometry material in Math 11H (3H). This is a development version of the text that should be considered a work-in-progress. This review guide and other review material are developed by the CHSN Review Project. Copyright © 2008-2009 Dara Adib. This is a freely licensed work, as explained in the Definition of Free Cultural Works (freedomdefined.org). It is licensed under the Creative Commons Attribution- Share Alike 3.0 United States License. To view a copy of this license, visit http://creativecommons. org/licenses/by-sa/3.0/us/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. This review guide is provided “as is” without warranty of any kind, either expressed or implied. You should not assume that this review guide is error-free or that it will be suitable for the particular purpose which you have in mind when using it. In no event shall the CHSN Review Project be liable for any special, incidental, indirect or consequential damages of any kind, or any damages whatsoever, including, without limitation, those resulting from loss of use, data or profits, whether or not advised of the possibility of damage, and on any theory of liability, arising out of or in connection with the use or performance of this review guide or other documents which are referenced by or linked to in this review guide. 1 Angles and Sectors of Circles Radians π . • To convert degrees to radians, multiply by 180 • To convert radians to degrees, multiply by 180 π . • One radian equals 180 ◦ π or approximately 57.296 . Coterminal Angles coterminal angles different angles that have the same initial and terminal ray The differences between coterminal angles are multiples of 360◦ . Examples: 60◦ , −300◦ , 420◦ Sectors sector part of a circle formed by two radii and an arc s = rθ (arc length = radius × angle) The angle must be represented in radians. apparent size the angle that an object subtends at one’s eyes; this explains why an object appears to be smaller when farther away Trigonometric Functions Reciprocal Functions The following are reciprocal functions. 1 • csc x = sin x 1 • sec x = cos x 1 • cot x = tan x As a result, the following are true as well. 1 • sin x = csc x 1 • cos x = sec x 1 • tan x = cot x 2 θ (degrees) θ (radians) Point sin θ cos θ tan θ 0◦ 0 (1, 0) 0 1 0 π 90◦ (0, 1) 1 0 undefined 2 180◦ π (−1, 0) 0 −1 0 3π 270◦ (0, −1) −1 0 undefined 2 θ (degrees) θ (radians) Point cot θ sec θ csc θ 0◦ 0 (1, 0) undefined 1 undefined π 90◦ (0, 1) 0 undefined 1 2 180◦ π (−1, 0) undefined −1 undefined 3π 270◦ (0, −1) 0 undefined −1 2 Table 1: quadrantal angles Cofunctions The following are cofunctions. • sin x = cos(90 − x) • tan x = cot(90 − x) • sec x = csc(90 − x) As a result, the following are true as well. • cos x = sin(90 − x) • cot x = tan(90 − x) • csc x = sec(90 − x) Special Angles Quadrantal Angles quadrantal angles angles that have a terminal side coinciding with a coordinate axis The value of the trigonometric function (i.e. sine, cosine, tangent) is determined by the coordinates of the points on a unit circle. By definition, the point (x, y) on a unit circle corresponds to (cos θ, sin θ). As a result, quadrantal angles can be determined easily. Table 1 lists quadrantal angles and the values sin θ of trigonometric functions. Since tan θ = cos θ , the tangent function of an angle can be found by dividing the sine function of the angle by the cosine function of the angle. 3 Quadrant Reference Angle (β) I β=θ II β = 180◦ − θ III β = θ − 180◦ IV β = 360◦ − θ Table 2: reference angles Angles greater than 90◦ Angles greater than 90◦ may be in different quadrants. The trigonometric functions vary over whether the trigonometric function of a certain angle is positive or negative. It is useful to remember the mnemonic device “All Students Take Calculus.” • All trigonometric functions are positive in the first quadrant • Sine and cosecant are positive in the second quadrant • Tangent and cotangent are positive in the third quadrant • Cosine and secant are positive in the fourth quadrant • The remaining trigonometric functions in each quadrant are negative. Reference Angles The use of reference angles is a way to simplify the calculation of the values of trigonometric functions in different quadrants. Refer to Table 2, where θ is the angle (in degrees) and β is the reference angle. Sum and Difference Formulas Sum Formulas Sine sin(A + B) = sin A cos B + cos A sin B Cosine cos(A + B) = cos A cos B − sin A sin B Tangent The derivation for the tangent of the sum of two angles follows. tan(A + B) sin(A + B) cos(A + B) 4 sin A cos B + cos A sin B cos A cos B − sin A sin B sin A cos B cos A sin B + cos A cos B cos A cos B cos A cos B sin A sin B − cos A cos B cos A cos B tan A + tan B 1 − tan A tan B Difference Formulas Sine sin(A − B) = sin A cos B − cos A sin B Cosine cos(A − B) = cos A cos B + sin A sin B Tangent The derivation for the tangent of the difference of two angles follows. tan(A − B) sin(A − B) cos(A − B) sin A cos B − cos A sin B cos A cos B + sin A sin B sin A cos B cos A sin B − cos A cos B cos A cos B cos A cos B sin A sin B + cos A cos B cos A cos B tan A − tan B 1 + tan A tan B Identities Pythagorean Identities 1. sin2 θ + cos2 θ = 1 2. 1 + tan2 θ = sec2 θ 3. 1 + cot2 θ = csc2 θ 5 Quotient Identities sin θ 1. tan θ = cos θ cos θ 2. cot θ = sin θ Graphing Trigonometric Functions General Form The general form of a trigonometric function is one of the following (sine or cosine). y = a sin b(x ± c) ± d y = a cos b(x ± c) ± d In other words, it can be understood to mean the following. y = amplitude × sin [frequency (x − horizontal translation)] + vertical translation y = amplitude × cos [frequency (x − horizontal translation)] + vertical translation A negative amplitude means a reflection over the x-axis. The vertical translation may be kept in front only to prevent the ambiguity that the number may be part of the trigonometric function. Frequency and Period The following statements are true regarding the relationship between frequency and period. 2π frequency = period 2π period = frequency Sine and Cosine The sine function follows the form zero, maximum, zero, minimum, before any translation. The cosine function follows the form maximum, zero, minimum, zero, before any translation. As a result, sine and cosine are horizontal translations of each other. Double and Half Angle Formulas Double Angle Formulas The following are determined by plugging in an angle twice into the sum formulas. 6 Sine sin 2A = 2 sin A cos A Cosine cos 2A = cos2 A − sin2 A More Cosine Formulas However, more formulas for the double of an angle with cosine can be determined since sin2 A + cos2 A = 1 (first Pythagorean identity). cos 2A = 1 − 2 sin2 A cos 2A = 2 cos2 A − 1 Tangent 2 tan A tan 2A = 1 − tan2 A Half Angle Formulas Sine r 1 1 − cos A sin A = ± 2 2 Cosine r 1 1 + cos A cos A = ± 2 2 Tangent r 1 1 − cos A tan A = ± 2 1 + cos A 7