You are given the following information aboutĀ Īø
sinĪø=23,Ļ2<Īø<Ļ
What areĀ cosĪøĀ andĀ tanĪø?
Trigonometric Identities
You can use the Pythagorean, Tangent andĀ Reciprocal IdentitiesĀ to find all sixĀ trigonometricĀ values for certainĀ angles. Letās walk through a fewĀ problemsĀ so that you understand how to do this.
Let's solve the following problems usingĀ trigonometricĀ identities.
Given thatĀ cosĪø=35Ā andĀ 0<Īø<Ļ2, findĀ sinĪø.
Use theĀ Pythagorean IdentityĀ to findĀ sinĪø.
sin2Īø+cos2Īøsin2Īø+(35)2sin2Īøsin2ĪøsinĪø=1=1=1ā925=1625=Ā±45
BecauseĀ ĪøĀ is in the first quadrant, we know that sine will be positive.Ā sinĪø=45
FindĀ tanĪøĀ from #1 above.
Use the Tangent Identity to findĀ tanĪø.
tanĪø=sinĪøcosĪø=4535=43
Find the other threeĀ trigonometricĀ functionsĀ ofĀ ĪøĀ from #1.
To find secant, cosecant, and cotangent use theĀ Reciprocal Identities.
cscĪø=1sinĪø=145=54secĪø=1cosĪø=135=53cotĪø=1tanĪø=143=34
ExamplesExample 1
Earlier, you were askedĀ to findĀ cosĪøĀ andĀ tanĪøĀ of Ā sinĪø=23,Ļ2<Īø<Ļ.Ā
First, use theĀ Pythagorean IdentityĀ to findĀ cosĪø.
sin2Īø+cos2Īø(23)2+cos2Īø=1cos2Īøcos2ĪøcosĪø=1=1ā49=59=Ā±5ā3
However, becauseĀ ĪøĀ is restricted to the second quadrant, the cosine must be negative. Therefore,Ā cosĪø=ā5ā3.
Now use the Tangent Identity to findĀ tanĪø.
tanĪø=sinĪøcosĪø=23ā5ā3=ā25ā=ā25ā5
Find the values of the other five trigonometricĀ functions.
Example 2
tanĪø=ā512,Ļ2<Īø<Ļ
First, we know thatĀ ĪøĀ is in the second quadrant, making sine positive and cosine negative. For this problem, we will use theĀ Pythagorean IdentityĀ 1+tan2Īø=sec2ĪøĀ to find secant.
1+(ā512)21+25144169144Ā±1312ā1312=sec2Īø=sec2Īø=sec2Īø=secĪø=secĪø
IfĀ secĪø=ā1312, thenĀ cosĪø=ā1213.Ā sinĪø=513Ā because the numerator value of tangent is the sine and it has the same denominator value as cosine.Ā cscĪø=135Ā andĀ cotĪø=ā125Ā from theĀ Reciprocal Identities.
Example 3
cscĪø=ā8,Ļ<Īø<3Ļ2
ĪøĀ is in the third quadrant, so both sine and cosine are negative. The reciprocal ofĀ cscĪø=ā8, will give usĀ sinĪø=ā18. Now, use the Pythagorean IdentityĀ sin2Īø+cos2Īø=1Ā to find cosine.
(ā18)2+cos2Īøcos2Īøcos2ĪøcosĪøcosĪø=1=1ā164=6364=Ā±37ā8=ā37ā8
secĪø=ā837ā=ā87ā21,tanĪø=137ā=7ā21,Ā andĀ cotĪø=37ā
ReviewIn which quadrants is the sine value positive? Negative?In which quadrants is the cosine value positive? Negative?In which quadrants is the tangent value positive? Negative?
Find the values of the other five trigonometricĀ functionsĀ ofĀ Īø.
sinĪø=817,0<Īø<Ļ2cosĪø=ā56,Ļ2<Īø<ĻtanĪø=3ā4,0<Īø<Ļ2secĪø=ā419,Ļ<Īø<3Ļ2sinĪø=ā1114,3Ļ2<Īø<2ĻcosĪø=2ā2,0<Īø<Ļ2cotĪø=5ā,Ļ<Īø<3Ļ2cscĪø=4,Ļ2<Īø<ĻtanĪø=ā710,3Ļ2<Īø<2ĻAside from using the identities, how else can you find the values of the other five trigonometric functions?Given thatĀ cosĪø=611Ā andĀ ĪøĀ is in theĀ 2ndĀ quadrant, what isĀ sin(āĪø)?Given thatĀ tanĪø=ā58Ā andĀ ĪøĀ is in theĀ 4thĀ quadrant, what isĀ sec(āĪø)?