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© T Madas Enlargement Scale Factor In Proportion Constant Ratio 8 4 3 6 © T Madas Enlargement Scale Factor In Proportion Constant Ratio 8 4 3 6 © T Madas © T Madas Hypotenuse Lies opposite the right angle The longest side of a right angled triangle © T Madas “thita” is a Greek letter we use to mark angles Opposite Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ The hypotenuse is always the same but the other 2 sides change if θ changes θ Adjacent © T Madas “thita” is a Greek letter we use to mark angles θ Adjacent Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ The hypotenuse is always the same but the other 2 sides change if θ changes Opposite © T Madas “thita” is a Greek letter we use to mark angles Opposite Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ The hypotenuse is always the same but the other 2 sides change if θ changes θ Adjacent © T Madas “thita” is a Greek letter we use to mark angles θ Adjacent Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ The hypotenuse is always the same but the other 2 sides change if θ changes Opposite © T Madas “thita” is a Greek letter we use to mark angles Opposite Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ The hypotenuse is always the same but the other 2 sides change if θ changes θ Adjacent © T Madas “thita” is a Greek letter we use to mark angles θ Adjacent Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ The hypotenuse is always the same but the other 2 sides change if θ changes Opposite © T Madas “thita” is a Greek letter we use to mark angles Opposite Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ The hypotenuse is always the same but the other 2 sides change if θ changes θ Adjacent This will become important later © T Madas The Beginning of Trigonometry © T Madas Opposite = 2 3 Opposite = 8 Adjacent = 8 12 θ Adjacent = 12 © T Madas Opposite = 4 Opposite Adjacent = 4 6 = 2 3 Why is this angle also θ ? θ Adjacent = 6 θ © T Madas Opposite Opposite = 6 Adjacent = 6 9 = 2 3 θ Adjacent = 9 θ © T Madas Opposite = 5 Opposite Adjacent = 2 5 = 3 7.5 θ Adjacent = 7.5 θ © T Madas Opposite Opposite Adjacent θ θ = 2 3 θ θ Adjacent © T Madas For a given acute angle of a right angled triangle: Opposite Opposite Adjacent θ = constant tangent θ of θ θ θ Adjacent © T Madas For a given acute angle of a right angled triangle: Opposite Opposite Adjacent θ = θ tangent of θ θ θ θ Adjacent © T Madas Tangent Practice © T Madas What is the tangent of θ ? What is the tangent of a ? 3 a 5 4 θ Opposite tanθ = Adjacent = 3 = 0.75 4 Opposite tana = Adjacent = 4 ≈ 1.33 3 © T Madas What is the tangent of x ? 5 x 13 12 tanx = Opp = 12 = 2.4 Adj 5 © T Madas What is the tangent of y ? Opp tany = = 15 = 1.875 Adj 8 17 15 y 8 © T Madas What is the tangent of θ ? 25 θ 7 24 tanθ = Opp = 24 ≈ 2.43 Adj 7 © T Madas In every right angled triangle: Opposite side Adjacent side = constant Opposite side Adjacent side = tanθ Every acute angle θ has its tanθ (constant ratio) stored in your calculator © T Madas SHIFT ALPHA x! nPr ^ nCr a b/c (–) Rec( x c A : Pol( REPLAY ^ d/ ^ ^ x-1 MODE CLR x2 B . , ,, RCL ENG ^ log sin cos 3 x3 ex e ln ( ) ; X , Y M- M M+ OFF 8 9 DEL 4 5 6 x ÷ 1 2 3 + – Ran# π DRG› % 0 . Out NOW! tan 7 Rnd Calculators C sin-1 D cos-1 E tan-1 F hyp STO 10x ON EXP Ans AC = © T Madas SHIFT ALPHA x! nPr ^ nCr a b/c (–) Rec( x c A : Pol( REPLAY ^ d/ ^ ^ x-1 MODE CLR x2 B . , ,, RCL ENG ^ log sin cos 3 x3 ex e ln C sin-1 D cos-1 E tan-1 F hyp STO 10x ON ( ) ; X , Y tan M- M M+ OFF 7 8 9 DEL 4 5 6 x ÷ 1 2 3 + – Ran# π DRG› % Rnd 0 . EXP Ans Find the tangent button in your calculator AC = © T Madas ta 30 0.5773502 n 69 SHIFT ALPHA x! nPr ^ ^ nCr a b/c (–) Rec( x c A : Pol( REPLAY ^ d/ MODE CLR ^ x-1 x2 B . , ,, RCL ENG 10x ^ log sin cos x3 ex e ln ( ) ; X , Y tan M- M M+ OFF 9 DEL 4 5 6 x ÷ 1 2 3 + – Ran# π DRG› % . = 3 8 0 0 ON 7 Rnd 3 C sin-1 D cos-1 E tan-1 F hyp STO tan EXP Ans AC = © T Madas ta 64 2.0503038 n 42 SHIFT ALPHA x! nPr ^ ^ nCr a b/c (–) Rec( x c A : Pol( REPLAY ^ d/ MODE CLR ^ x-1 x2 B . , ,, RCL ENG 10x ^ log sin cos x3 ex e ln ( ) ; X , Y tan M- M M+ OFF 9 DEL 4 5 6 x ÷ 1 2 3 + – Ran# π DRG› % . = 3 8 0 4 ON 7 Rnd 6 C sin-1 D cos-1 E tan-1 F hyp STO tan EXP Ans AC = © T Madas ta 29 0.5543090 n 51 SHIFT ALPHA x! nPr ^ ^ nCr a b/c (–) Rec( x c A : Pol( REPLAY ^ d/ MODE CLR ^ x-1 x2 B . , ,, RCL ENG 10x ^ log sin cos x3 ex e ln ( ) ; X , Y tan M- M M+ OFF 9 DEL 4 5 6 x ÷ 1 2 3 + – Ran# π DRG› % . = 3 8 0 9 ON 7 Rnd 2 C sin-1 D cos-1 E tan-1 F hyp STO tan EXP Ans AC = © T Madas tan 0 5 . -1 26.565051 18 SHIFT ALPHA x! nPr ^ ^ nCr a b/c (–) Rec( x c A : Pol( REPLAY ^ d/ MODE CLR ^ x-1 x2 B . , ,, RCL ENG 10x ^ log sin cos ( ) ; X , Y x3 ex e ln tan M- M M+ OFF 8 9 DEL 4 5 6 x ÷ 1 2 3 + – Ran# π DRG› % 0 . EXP Ans 0 . 5 = 3 7 Rnd tan ON C sin-1 D cos-1 E tan-1 F hyp STO shift AC You can use the calculator to work backwards from a tangent to an angle This is known as the inverse of the tangent It is written as : tan-1 Which acute angle in a right angled triangle has tangent equal to ½ ? = © T Madas tan 4. 7 01 -1 76.315522 16 SHIFT ALPHA x! nPr ^ nCr x c a b/c A (–) Rec( ^ d/ : Pol( REPLAY ^ x-1 MODE CLR ^ x2 B . , ,, STO RCL ENG 10x ^ log sin cos ON 3 x3 ex e ln C sin-1 D cos-1 E tan-1 F hyp ( ) ; X , Y tan M- M M+ OFF 7 8 9 DEL 4 5 6 x ÷ 1 2 3 + – Ran# π DRG› % Rnd 0 . EXP Ans tan shift AC 4 . 0 1 7 = You can use the calculator to work backwards from a tangent to an angle This is known as the inverse of the tangent It is written as : tan-1 Which acute angle in a right angled triangle has tangent equal to 4.017 ? = © T Madas tan 0 533 . -1 28.057615 73 SHIFT ALPHA x! nPr ^ nCr x c a b/c A (–) Rec( ^ d/ : Pol( REPLAY ^ x-1 MODE CLR ^ x2 B . , ,, RCL ENG ^ log sin cos ON 3 x3 ex e ln C sin-1 D cos-1 E tan-1 F hyp STO 10x ( ) ; X , Y tan M- M M+ OFF 7 8 9 DEL 4 5 6 x ÷ 1 2 3 + – Ran# π DRG› % Rnd 0 . EXP Ans tan shift AC 0 . 5 3 3 = You can use the calculator to work backwards from a tangent to an angle This is known as the inverse of the tangent It is written as : tan-1 Which acute angle in a right angled triangle has tangent equal to 0.533 ? = © T Madas © T Madas