Trigonometry is the study and solution ofTriangles. Solving a triangle means finding the value of each of its sides and angles.The following terminology and tactics will be important in the solving of triangles. Pythagorean Theorem (a2+b2=c2). Only for right angle triangles Sine (sin), Cosecant (csc or sin-1) Cosine (cos), Secant (sec or cos-1) Tangent (tan), Cotangent (cot or tan-1) Right/Oblique triangle
us e Since a triangle has three ten sides, there are six ways to adjacent o divide the lengths of the hyp sides Each of these six ratios has a name (and an abbreviation) The ratios depend on the Three ratios are most used: shape of the triangle (the opposite sine = sin = opp / hyp cosine = cos = adj / hyp angles) but not on the size tangent = tan = opp / adj e The other three ratios are nus ote adjacent cosecant= cosec= hyp/ opp hyp secant= sec= hyp/ adj cotangent = cot = adj/opp opposite
THE SIDE OPPOSITE TO THE ANGLE angle opposite opposite oppositeangle angle angle opposite OP PO SIT E SID E
THE SIDE ADJACENT TO THE ANGLE angleangle angleadjacent angle t nec a da t nec a da j j ADJACENT
THE LONGEST SIDE se enuhy pot e h yp e nus ote nu hy pot se hyp o te n use HY PO TE NU SE
THREE TYPES TRIGONOMETRICRATIOS There are 3 kinds of trigonometric ratios we will learn. sine ratio cosine ratio tangent ratio
sine ratio θFor any right-angled triangle Opposite side Sinθ = hypotenuses
θFor any right-angled triangle Adjacent Side Cosθ = hypotenuses
θFor any right-angled triangle Opposite Side tanθ = Adjacent Side
Reciprocal Identities 1 1 1 cot θ = secθ = cscθ = tan θ cosθ sin θQuotient Identities sin θ cosθ tan θ = cot θ = cosθ sin θPythagorean Identities sin θ + cos θ = 1 tan θ + 1 = sec θ 1 + cot θ = csc θ 2 2 2 2 2 2Negative-Number Identities sin( −θ ) = − sin θ cos( −θ ) = cosθ tan( −θ ) = − tan θ
Work with one side at a time. We want both sides to be exactly the same. Start with either side Use algebraic manipulations and/or the basic trigonometric identities until you have the same expression as on the other side.
cot x sin x = cos xLHS = cot x sin x and RHS = cos x cos x = ⋅ sin x sin x = cos x Since both sides are the same, the identity is verified.
Start with the more complicated side Try substituting basic identities (changing all functions to be in terms of sine and cosine may make things easier) Try algebra: factor, multiply, add, simplify, split up fractions If you’re really stuck make sure to: Change everything on both sides to sine and cosine.