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THE UNIVERSITY OF AKRON Theoretical and Applied Mathematics Instructions: Click on the Begin button to view the first randomly selected card. Click on FS to view the cards in full screen mode (works only outside a web browser). The Home button on the first page goes to the WebTrig home page; otherwise, the Home button returns to this page. The Close button closes the document (use outside a web browser). c 2003 [email protected] Last Revision Date: April 12, 2003 AcroTEX eDucation Bundle Katie Jones and Tom Price WebTrig Flash Cards Flash Cards Trigonometric Identities Begin FS Close Home Version 1.0 AcroTEX eDucation Bundle tan2 +1 = sec2 t. WebTrig Flash Cards Verify that Hint Soln Next Home AcroTEX eDucation Bundle 2 sin t + cos2 t = 2. WebTrig Flash Cards Solve the equation Hint Soln Next Home AcroTEX eDucation Bundle WebTrig Flash Cards Show that π − α = tan α. cot 2 Hint Soln Next Home AcroTEX eDucation Bundle WebTrig Flash Cards Without a calculator determine the value of 7π sin . 12 Hint Soln Next Home AcroTEX eDucation Bundle cos 285◦. WebTrig Flash Cards Without a calculator determine the value of Hint Soln Next Home AcroTEX eDucation Bundle tan 75◦. WebTrig Flash Cards Without a calculator determine the value of Hint Soln Next Home AcroTEX eDucation Bundle WebTrig Flash Cards Without a calculator determine the value of 5π tan . 12 Hint Soln Next Home AcroTEX eDucation Bundle WebTrig Flash Cards Without a calculator determine the value of 13π sin . 12 Hint Soln Next Home AcroTEX eDucation Bundle cos(2α) = 2 cos2 α − 1. WebTrig Flash Cards Verify the double-angle formula Hint Soln Next Home AcroTEX eDucation Bundle WebTrig Flash Cards Verify the half-angle formula for the sine function: α 1 − cos α sin = ± . 2 2 Hint Soln Next Home AcroTEX eDucation Bundle sin 15◦. WebTrig Flash Cards Without a calculator determine the value of Hint Soln Next Home AcroTEX eDucation Bundle WebTrig Flash Cards Without a calculator determine the value of 8π cos . 3 Hint Soln Next Home AcroTEX eDucation Bundle WebTrig Flash Cards Verify the cofunction identity π − α = cos α sin 2 π for the value of sin − 6 . Hint Soln Next Home AcroTEX eDucation Bundle cos α sin β 1 = [sin (α + β) − sin (α − β)] 2 for the values π 5π α = and β = . 3 6 WebTrig Flash Cards Verify the product formula Hint Soln Next Home α+β α−β = 2 cos cos 2 2 for the values π 2π α = and β = . 3 3 AcroTEX eDucation Bundle cos α + cos β WebTrig Flash Cards Verify the factor formula Hint Soln Next Home AcroTEX eDucation Bundle WebTrig Flash Cards Simplify the expression cos2 t . 1 − sin t Hint Soln Next Home AcroTEX eDucation Bundle sec t cot t = csc t. WebTrig Flash Cards Show that Hint Soln Next Home AcroTEX eDucation Bundle sin t cot t = cos t. WebTrig Flash Cards Show that Hint Soln Next Home AcroTEX eDucation Bundle tan (−t) = − tan t. WebTrig Flash Cards Verify that Hint Soln Next Home AcroTEX eDucation Bundle cos 3a = 4 cos3 a − 3 cos a. WebTrig Flash Cards Show that Hint Soln Next Home HINT use the Pythagorean identity involving sine and cosine. AcroTEX eDucation Bundle tan2 +1 = sec2 t WebTrig Flash Cards To verify that Hint Soln Next Home Solution: Divide both sides of the Pythagorean identity sin2 t + cos2 t = 1 AcroTEX eDucation Bundle =⇒ 1 sin2 t +1= cos2 t cos2 t tan2 +1 = sec2 t. WebTrig Flash Cards by cos2 t to obtain Hint Soln Next Home HINT the pythagorean identity sin2 t + cos2 t = 1. AcroTEX eDucation Bundle 2 sin t + cos2 t = 2 WebTrig Flash Cards To solve the equation Hint Soln Next Home Answer: t = π 2 + 2πk =⇒ 2 sin t + 1 − sin2 t = 2 =⇒ 2 sin t − 1 − sin2 t = 0 =⇒ sin2 t − 2 sin t + 1 = 0 =⇒ (sin t − 1) = 0 sin t = 1 =⇒ 2 The sine function equals 1 whenever t = integer k. π 2 AcroTEX eDucation Bundle 2 sin t + cos2 t = 2 WebTrig Flash Cards Solution: We know that sin2 t + cos2 t = 1, so cos2 t = 1 − sin2 t. Hence, + 2πk for some Hint Soln Next Home HINT AcroTEX eDucation Bundle WebTrig Flash Cards To show that π − α = tan α cot 2 use the cofunction identity involving tangent. Hint Soln Next Home AcroTEX eDucation Bundle WebTrig Flash Cards Solution: Using the definition of the cotangent function and a similar cofunction identity for the tangent function, we have π 1 1 cot( − α) = = = tan α. 2 tan( π2 − α) cot α Hint Soln Next Home HINT = π 3 + π4 . AcroTEX eDucation Bundle 7π 12 WebTrig Flash Cards To find sin 7π , notice that 12 Hint Soln Next Home Answer: sin 7π 12 √ = √ 6+ 2 4 = = √ √ √ 3 2 2 2 · 2 + 2 √ √ 6 2 4 + 4 √ √ 6+ 2 . 4 · 1 2 AcroTEX eDucation Bundle = WebTrig Flash Cards π π Solution: Notice 7π 12 = 3 + 4 , which are both basic angles we know from the unit circle. So, π π sin 7π 12 = sin 3 + 4 = sin π3 cos π4 + sin π4 cos π3 Hint Soln Next Home HINT cos (a + b) = cos a cos b − sin a sin b. AcroTEX eDucation Bundle cos 285◦ use this the sum formula for the cosine function WebTrig Flash Cards To find Hint Soln Next Home ◦ Answer: cos 285 = √ √ − 2− 6 4 = cos 240 cos 45 − sin 240 sin 45 √ √ √ = − 12 · 22 + − 23 · 22 √ √ = − 42 + − 46 = √ √ − 2− 6 . 4 AcroTEX eDucation Bundle cos 285◦ = cos (240◦ + 45◦ ) WebTrig Flash Cards Solution: Notice 285 = 240 + 45. So, Hint Soln Next Home HINT AcroTEX eDucation Bundle tan 75◦ a the difference formula for tangent. WebTrig Flash Cards To find Hint Soln Next Home ◦ Answer: tan 75 = √ −1−√ 3 1− 3 AcroTEX eDucation Bundle tan 75◦ = tan (135◦ − 60◦ ) tan 135 − tan 60 = 1 + tan 135 tan 60 √ −1 − 3 √ = 1 + (−1) 3 √ −1 − 3 √ . = 1− 3 WebTrig Flash Cards Solution: Notice 75 = 135 − 60. So, Hint Soln Next Home HINT AcroTEX eDucation Bundle 5π 12 find two numbers whose sum or difference and use the sum formula for tangent is 5π 12 function. tan WebTrig Flash Cards To find Hint Soln Next Home Answer: tan 5π 12 = √ 1− √ 3 3−1 WebTrig Flash Cards AcroTEX eDucation Bundle 5π π π 12 = 6 + 4 . So, π π tan 5π 12 = tan 6 + 4 tan π6 + tan π4 = 1 − tan π6 tan π4 √1 − 1 3 = 1 − √13 · 1 Solution: Notice √ 1− 3 =√ . 3−1 Hint Soln Next Home HINT AcroTEX eDucation Bundle 13π 12 use the difference formula for the sine function: sin (a − b) = sin a cos b − sin b cos a. sin WebTrig Flash Cards To find Hint Soln Next Home Answer: sin 13π 12 π = 4π 3 − 4 . So, 4π π = sin 3 − 4 cos π − sin π cos 4π = sin 4π √3 √ 4 √ 4 3 = − 23 22 − 22 − 12 = = √ √ − 6 2 + 4 4 √ √ − 6+ 2 . 4 AcroTEX eDucation Bundle sin 13π 12 = √ √ − 6+ 2 4 WebTrig Flash Cards Solution: Notice 13π 12 Hint Soln Next Home HINT use the fact that 2a = a + a. AcroTEX eDucation Bundle cos(2α) = 2 cos2 α − 1 WebTrig Flash Cards To verify the double-angle formula Hint Soln Next Home Solution: Since 2a = a + a, use the cosine formula for the sum of two angles and a pythagorean identity to solve = cos2 α − (1 − cos2 α) = 2 cos2 α − 1. AcroTEX eDucation Bundle = cos2 α − sin2 α WebTrig Flash Cards cos(2α) = cos(α + α) = cos α cos α − sin α sin α Hint Soln Next Home HINT AcroTEX eDucation Bundle cos(2t) = 1 − 2 sin2 t. WebTrig Flash Cards To verify the half-angle formula for the sine function: 1 − cos α α , sin = ± 2 2 start with the double angle identity Hint Soln Next Home Solution: By the double angle formula for the cosine function we have 2 sin2 t = 1 − cos(2t) Dividing this last equation through by 2 and then taking the square root of both sides yields 1 − cos(2t) . sin t = ± 2 Replacing t with α2 in this equation yields the desired halfangle formula 1 − cos α α . sin = ± 2 2 AcroTEX eDucation Bundle =⇒ WebTrig Flash Cards cos(2t) = 1 − 2 sin2 t Hint Soln Next Home HINT AcroTEX eDucation Bundle WebTrig Flash Cards To find sin 15◦ , notice that 30 15 = . 2 Hint Soln Next Home Answer: sin √ = √ 2− 3 2 AcroTEX eDucation Bundle ◦ Solution: Notice 15 = 30 2 , and 30 is a known basic angle. So, use the half angle formula for the sine function to solve. √ √ √ 3 ◦ 1− 2 1 − cos 30 2− 3 2− 3 = = = . = 2 2 4 2 WebTrig Flash Cards 30 ◦ 2 Hint Soln Next Home HINT AcroTEX eDucation Bundle WebTrig Flash Cards To find cos 8π , notice that 3 4π 8π =2· 3 3 and use the double angle formula for cosine. Hint Soln Next Home 1 Answer: cos 8π 3 = −2 AcroTEX eDucation Bundle WebTrig Flash Cards 4π Solution: Notice 8π 3 = 2 · 3 and use the double angle formula for cosine to obtain cos 2 · 4π = 2 cos2 4π 3 3 −1 2 1 =2 − −1 2 1 −1 =2 4 1 =− . 2 Hint Soln Next Home HINT AcroTEX eDucation Bundle WebTrig Flash Cards To verify the cofunction identity π − α = cos α sin 2 for the value of sin − π6 , first set up an equation to find α. Hint Soln Next Home AcroTEX eDucation Bundle WebTrig Flash Cards Solution: First find α by solving π π − = −α 6 2 π π 2π =⇒ α= + = . 2 6 3 Now, evaluate = − 12 = cos 2π sin − π6 = sin π2 − 2π 3 3 . Hint Soln Next Home HINT AcroTEX eDucation Bundle WebTrig Flash Cards To verify the product formula 1 cos α sin β = [sin (α + β) − sin (α − β)] 2 for the values 5π π . α = and β = 3 6 use . sin (α + β) = sin π3 + 5π 6 Hint Soln Next Home Answer: Both yield the value 1 4 Also, 1 [sin (α + β) − sin (α − β)] = 2 = 1 2 · 1 2 = 14 . π 5π sin 3 + 6 − sin π3 − 7π π 1 2 sin 6 − sin − 2 = 12 − 12 − (−1) = 14 . 1 2 5π 6 AcroTEX eDucation Bundle cos α sin β = cos π3 sin 5π 6 = WebTrig Flash Cards Solution: In this case, Hint Soln Next Home HINT = 2 cos α+β 2 for the values α = = − π6 . that α−β 2 π 3 cos and β = α−β 2 2π , 3 observe AcroTEX eDucation Bundle cos α + cos β WebTrig Flash Cards To verify the factor formula Hint Soln Next Home Answer: Both yield the value 0 WebTrig Flash Cards Solution: In this case, Also, π 2π π 2π + 3 − 3 α+β α−β 2 cos cos = 2 cos 3 cos 3 2 2 2 2 π −π = 2 cos 2 cos 6 √ = 2 (0) 23 = 0. AcroTEX eDucation Bundle 2π 1 1 π = + − = 0. cos α + cos β = cos + cos 3 3 2 2 This means that both sides of the given equation are zero which establishes the desired equality. Hint Soln Next Home HINT AcroTEX eDucation Bundle cos2 t = 1 − sin2 t. WebTrig Flash Cards To simplify the expression cos2 t 1 − sin t use the pythagorean identity Hint Soln Next Home AcroTEX eDucation Bundle WebTrig Flash Cards Solution: Use the Pythagorean identities and factoring to establish cos2 t 1 − sin2 t = 1 − sin t 1 − sin t (1 − sin t) (1 + sin t) =⇒ = 1 − sin t =⇒ = 1 + sin t. Hint Soln Next Home HINT use the basic reciprocal identities 1 cos t . sec t = and cot t = cos t sin t AcroTEX eDucation Bundle sec t cot t = csc t WebTrig Flash Cards To show that Hint Soln Next Home AcroTEX eDucation Bundle WebTrig Flash Cards Solution: Use the reciprocal identities to solve. cos t 1 · sec t cot t = cos t sin t 1 = sin t = csc t. Hint Soln Next Home HINT use the basic reciprocal identities. AcroTEX eDucation Bundle sin t cot t = cos t WebTrig Flash Cards To show that Hint Soln Next Home AcroTEX eDucation Bundle WebTrig Flash Cards Solution: Use the reciprocal identities to solve. cos t sin t cot t = sin t · sin t = cos t. Hint Soln Next Home HINT remember that the sine function is odd and cosine is even. That is, sin (−t) = − sin t and cos (−t) = cos t. AcroTEX eDucation Bundle tan (−t) = − tan t, WebTrig Flash Cards To verify that Hint Soln Next Home Solution: Use the symmetry identities to solve, remembering that the sine function is odd and cosine is even. WebTrig Flash Cards sin (−t) cos (−t) − sin t = cos t = − tan t. tan (−t) = AcroTEX eDucation Bundle Hint Soln Next Home HINT use the fact that 3a = 2a + a. AcroTEX eDucation Bundle cos 3a = 4 cos3 a − 3 cos a WebTrig Flash Cards To show that Hint Soln Next Home = 2 cos3 a − cos a − 2 cos a sin2 a = 2 cos3 a − cos a − 2 cos a 1 − cos2 a = 2 cos3 a − cos a − 2 cos a + 2 cos3 a = 4 cos3 a − 3 cos a. AcroTEX eDucation Bundle cos 3a = cos (2a + a) = cos 2a cos a − sin 2a sin a = 2 cos2 a − 1 cos a − 2 sin a cos a sin a WebTrig Flash Cards Solution: Use the fact that 3a = 2a + a and the sum, difference, Pythagorean, and double angle formulas to arrive at the following: Hint Soln Next Home