Review of Trigonometry for Calculus 1 iversit Un as DEO PAT- ET RIÆ S a s s si ka n Review of Trigonometry for Calculus tchewane 2002 Doug MacLean “Trigon” =triangle +“metry”=measurement =Trigonometry so Trigonometry got its name as the science of measuring triangles. When one first meets the trigonometric functions, they are presented in the context of ratios of sides of a right-angled triangle, where a2 + o2 = h2: We have o opposite h sin α = = , h hypotenuse o a adjacent α cos α = = , and h hypotenuse a o opposite tan α = = , a adjacent which is often remembered with the “sohcahtoa” rule. If h = 1, we have sin α = o, cos α = a,so(sin α)2 + (cos α)2 = 1, so we know that the point (cos α, sin α) lies on the unit circle. 2 Review of Trigonometry for Calculus iversit Un as DEO PAT- ET RIÆ S a s s si ka n The Unit Circle: tchewane 2002 Doug MacLean In Calculus, most references to the trigonometric functions are based on the unit circle, x2 + y2 = 1. Points on this circle determine angles measured from the point (0, 1) on the x-axis, where the counter- clockwise direction is considered to be positive. Units of Angular Measurement The most natural unit of measurement for angles in Geometry is the right angle . The revolution is used in the study of rotary motion, and is what the “r” stands for in “rpm”s. The degree , 1/90 of a right angle, was probably first adopted for navigational purposes. The mil , 1/1600 of a right angle, is used by the military. However, the basic unit of measurement for angles in Calculus is the radian . Definition: A radian is the angle subtended by a circular arc on a circle whose length equals the radius of the circle. Thus, on the unit circle an angle whose size is one radian subtends a circular arc on the unit circle whose length is exactly one. y 1 radian x (0,0) (1,0) Figure 1. Review of Trigonometry for Calculus 3 iversit Un as DEO PAT- ET RIÆ S a s s si ka n Radian measure and degrees tchewane 2002 Doug MacLean Since the circumference of a circle is 2π times its radius, we have 2π radians = 360◦ = 4 right angles, so 360 ◦ 180 ◦ 4 right angles 2 1 radian = = = = right angles 2π π 2π π or 2π π 1 1◦ = radians = radians = right angles 360 180 90 In high school trigonometry, the trigonometric functions are used to solve problems concerning triangles and related geometric figures. In the Calculus, the trigonometric functions are used in the analysis of rotating bodies. It turns out that the degree, the unit of measurement of angles adopted by the Babylonians over 4,000 years ago, is not particularly well adapted to the analysis of jet engines, radar systems and CAT scanners. The radian is, because The sine and cosine functions live on the unit circle! If θ is a number, then cos θ and sin θ are defined to be the x- and y- coordinate, respectively, of the point on the unit circle obtained by measuring off the angle θ (in radians!) from the point (0, 1).Ifθ is positive, the angle is measured off in the counter-clockwise direction, and if θ is negative it is measured off in the clockwise direction. For an animated interactive look at these two functions, take a look at the applet Sine and Cosine Functions 4 Review of Trigonometry for Calculus iversit Un as DEO PAT- ET RIÆ S a s s si ka n tchewane 2002 Doug MacLean y θ sin θ x (0,0) cosθ Figure 2. The other trigonometric functions are now defined in terms of the first two: sin θ cos θ 1 1 tan θ = , cot θ = , sec θ = , csc θ = . cos θ sin θ cos θ sin θ Review of Trigonometry for Calculus 5 iversit Un as DEO PAT- ET RIÆ S a s s si Fundamental Angles of the First Quadrant: ka n tchewane 2002 Doug MacLean There are three acute angles for which the trigonometric function values are known and must be memorized by the student of Calculus. They are π π π (in radians) 6 , 4 , and 3 , (in degrees) 30◦,45◦, and 60◦, (in right angles) 1/3, 1/2, and 2/3. π In addition, the values of the trig functions for the angles 0 and 2 must be known. The following tables show how they may be easily constructed, if one can count from zero to four. The first table is a template, the second shows how it may be filled in, and the third contains the arithmetical simplifications of the values. Template: θ( ) π π π π radians 0 6 4 3 2 θ(degrees) 0 30 45 60 90 θ( ) 1 1 2 right angles 0 3 2 3 1 √ √ √ √ √ θ sin 2 2 2 2 2 √ √ √ √ √ θ cos 2 2 2 2 2 6 Review of Trigonometry for Calculus iversit Un as DEO PAT- ET RIÆ S a s s si Fill in the Blanks: ka n tchewane 2002 Doug MacLean θ( ) π π π π radians 0 6 4 3 2 θ(degrees) 0 30 45 60 90 θ( ) 1 1 2 right angles 0 3 2 3 1 √ √ √ √ √ θ 0 1 2 3 4 sin 2 2 2 2 2 √ √ √ √ √ θ 4 3 2 1 0 cos 2 2 2 2 2 Simplify the Arithmetic: θ( ) π π π π radians 0 6 4 3 2 θ(degrees) 0 30 45 60 90 θ( ) 1 1 2 right angles 0 3 2 3 1 √ √ θ 1 2 3 sin 0 2 2 2 1 √ √ θ 3 2 1 cos 1 2 2 2 0 Review of Trigonometry for Calculus 7 iversit Un as DEO PAT- ET RIÆ S a s Figure 3 shows these values on the first quadrant of the unit circle. s si ka n tchewane 2002 Doug MacLean y (0,1) — π ——13√ — ( , ——) 2 π 22 — √√22 3 ( —— , ) π 22— — 4 (√ —— 31 , ) π 22 — 6 0 (1,0) x Figure 3. 8 Review of Trigonometry for Calculus iversit Un as DEO PAT- ET RIÆ S a s s si ka n Moving Beyond the First Quadrant tchewane 2002 Doug MacLean These values may now be used to find the values of the trig functions at the other basic angles in the other three quadrants of unit circle. The same numerical values will appear, with the possible addition of minus signs. The following table gives the values, and the diagram displays them. The student should be able to reproduce them instantaneously! To do this, it will be necessary to be completely comfortable with the following identities, all of which are obvious from the symmetry of the unit circle: y π−θ θ π/2−θ x −θ θ+π Figure 4. Review of Trigonometry for Calculus 9 iversit Un as DEO PAT- ET RIÆ S a s s si ka n tchewane sin(π − θ) ≡ sin θ, cos(π − θ) ≡−cos θ 2002 Doug MacLean sin(θ + π) ≡−sin θ, cos(θ + π) ≡−cos θ sin(−θ) ≡−sin θ, cos(−θ) ≡ cos θ π π sin − θ ≡ cos θ, cos − θ ≡ sin θ 2 2 θ π π π π 2π 3π 5π π 7π 5π 4π 3π 5π 7π 11π 0 6 4 3 2 3 4 6 6 4 3 2 3 4 6 √ √ √ √ √ √ √ √ θ 1 2 3 3 2 1 − 1 − 2 − 3 − − 3 − 2 − 1 sin 0 2 2 2 1 2 2 2 0 2 2 2 1 2 2 2 √ √ √ √ √ √ √ √ θ 3 2 1 − 1 − 2 − 3 − − 3 − 2 − 1 1 2 3 cos 1 2 2 2 0 2 2 2 1 2 2 2 0 2 2 2 10 Review of Trigonometry for Calculus iversit Un as DEO PAT- y ET RIÆ S a s Figure 5 is left blank for the student to fill in: s si ka n (0,1) tchewane √— 2002 Doug MacLean (— , — ) π (— 13 , — ) — 22 2π 2 π √√—— (— , — ) — — ( — 22 , — ) 3 3 22 π π 3— — — (— , — ) 4 4 ( √ — 31 , — ) 22 π π 5— — 6 6 (-1,0) π 0 (1,0) x π π 7— 11— 6 6 (—— , ) π π (—— , ) 5— 7— 4 4 4π 5 π (—— , ) — — ( —— , ) 3 3π 3 (—— , ) — ( —— , ) 2 Figure 5. (0,-1) Review of Trigonometry for Calculus 11 iversit Un as DEO PAT- ET RIÆ S a s s si π π ka n Periodicity All six trig functions have period 2 , and two of them, tan and cot have period : tchewane 2002 Doug MacLean sin(θ + 2π) ≡ sin(θ) cos(θ + 2π) ≡ cos(θ) tan(θ + π) ≡ tan(θ) cot(θ + π) ≡ cot(θ) sec(θ + 2π) ≡ sec(θ) csc(θ + 2π) ≡ csc(θ) 12 Review of Trigonometry for Calculus iversit Un as DEO PAT- ET RIÆ S a s s si ka n Identities of the sine and cosine functions tchewane 2002 Doug MacLean The identity sin2 θ + cos2 θ ≡ 1 is obvious as a result of our use of the unit circle. It really should be written as (sin θ)2 + (cos θ)2 ≡ 1 but centuries of tradition have developed the confusing convention of writing sin2 θ for the square of sin θ. This identity leads to a number of other important identities and formulas: tan2 θ ≡ sec2 θ − 1 sec2 θ ≡ 1 + tan2 θ + 1 cot2 θ ≡ csc2 θ − 1 2 θ ≡ + 2 θ csc 1 cot sin θ =±1 − cos2 θ In addition to this fundamental knowledge, the student should be completely comfortable in deriving the trig identities which result from the fundamental identities for the sines and cosines of sums and differences of angles.