n Introduction to Trigonoetry First of ll, let s check out the right ngled tringle below. The LETTERS, B & C indicte the ngles nd the letters, b & c indicte the sides. c b It is iportnt to note tht side BC is opposite ngle. Tht is why BC =. Siilrly, C = b nd B = c. This nottion is used for esy reference. B C Now we hve to discuss the specil nes given to the sides of right tringles. These nes re used to define the foruls tht will be used. X Consider the tringle below: we cn use ny letters to denote the ngles. z ( opposite ) y ( hypotenuse ) The side opposite the right ngle is lwys lbelled the hypotenuse. If ngle Z is the reference ngle, then side XY = z is lbelled the opposite. Y ( djcent ) Z Side YZ = is lbelled the djcent. If ngle X is the reference ngle, then the hypotenuse reins the se, but side XY = z is lbelled the djcent nd side YZ = is lbelled the opposite. I ll let you sketch tht sitution to help you understnd. The right ngle = 90 cn never be used s the reference ngle for this definition. Now, we will get down to business. There re 3 priry trig rtios nd 3 reciprocl trig rtios. We will del with the priry trig rtios. Their nes re sine, cosine nd tngent. The rtios re lwys ssocited with n ngle.
Here re the bsic definitions: opposite djcent opposite sin cos tn hypotenuse hypotenuse djcent There is n crony to reeber the rtios...it is SOH CH TO. I ll let you figure tht out!!!!!! We will use these definitions shortly but let s look t clcultor use first. Use your scientific clcultor to clculte the vlue: sin 45. Just few coents...ll clcultors hve 3 settings: degrees, rdins nd grdients. They re just 3 wys to esure ngles. We will only del with degrees. There re ny types of clcultors...but bsiclly there re the old school ones nd those clled DL which ens Direct lgebric Logic. Clculte sin 45...ke sure your clcultor is on the degree setting. For old school clcultors...enter 45 then press sin. For DL press sin45. Both will give the nswer 0.7071 + severl ore digits. It s tht siple!!!! Now we will use right tringles to clculte unknown sides nd ngles... In ech tringle below, clculte the indicted side: First we ust choose the reference ngle...in this cse 61. Now, deterine the nes of the sides nd 23.5. 23.5 is djcent to 61 nd 23.5 is the hypotenuse. 61 This requires the trig rtio cosine...ch. djcent The rtio is cos hypotenuse Using siple substitution, we hve cos61 23.5. Now use clcultor to deterine the trig vlue... This gives us: 0.48480962 23.5 Net step: 0.48480962 23.5 11.39302608 nd finlly 11.4 ( hs the se ccurcy s side lengths in the question)
Now, let s do nother one. lso you cn use vribles other thn. 28 Reference ngle... 28 is the opposite side nd 13.84 is the djcent side. 13.84 tn 28 13.84 0.531709431 13.84 0.531709431 13.84 7.36 Now, we will look t one lst eple to deterine n unknown side. sin 42 42 0.669130606 0.669130606 using the Cross Product Rule. 0.669130606 11.8 The etr chllenge here ws tht the vrible ended up in the denointor. Using the Cross Product Rule tht difficulty ws overcoe. Now, let s look t soe right tringles tht only hve the right ngle given, long with ny 2 side lengths. Reeber: the Pythgoren Theore cn be used to clculte the 3 rd side length. We will focus on clculting the unknown vlues for the reining 2 ngles. There re severl possibilities...we will consider one of the. I will let you work on the others.
Clculte ngle in the eple below. 12.7 9.2 Reference ngle is...the ngle we wnt to clculte. 9.2 is the opposite side nd 12.7 is the hypotenuse...the trig rtio required is sine. oposite sin hypotenuse 9.2 sin 12.7 sin 0.724409448 Crry t lest 4 decil plces in this step. Fewer will ffect the ccurcy. Now, on your clcultor press 2 nd or INV or shift then press sin. These buttons re usully found on the top left. 46.41974309 46 round off to the nerest whole nuber. This gives you n intro to trig...soh CH TO...for right tringles. On the net pge, there re questions to prctice your Soh Ch To techniques.
Clculte the indicted quntity 1) 11.4 Clculte the vlue of. 27 2) In XYZ : X 90, Z 63 nd y 4.2. Clculte the vlue of z. 3) In BC : 57, B 90 nd b 19.4. Clculte the vlue of. 4) 6.1 Clculte the vlue of. 13.9 5) In EFG : F 90, f 9.3 nd g 2.5. Clculte the vlue of G. 6) In PQR : Q90, q 9.5 nd r 4.9. Clculte the vlue of P. 7) In BC : B90, C49 nd c 3.8. Clculte the vlue of b. 8) In XYZ : Y 90, Z 23 nd 8.1. Clculte the vlue of y. 9) In PQR : P50, R90 nd p 5.1. Clculte the vlue of q. nswers 1) 10.2 2) 8.2 3) 16.3 4) 24 5) 16 6) 59 7) 5.0 8) 8.8 9) 4.3