Introducing the Tangent © T Madas
8 4 3 6 Enlargement Scale Factor In Proportion Constant Ratio © T Madas
8 4 3 6 Enlargement Scale Factor In Proportion Constant Ratio © T Madas
© T Madas
Hypotenuse Hypotenuse Lies opposite the right angle The longest side of a right angled triangle Hypotenuse © T Madas
Hypotenuse Opposite θ Adjacent “thita” is a Greek letter we use to mark angles Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ Opposite Hypotenuse The hypotenuse is always the same but the other 2 sides change if θ changes θ Adjacent © T Madas
θ Hypotenuse Adjacent Opposite “thita” is a Greek letter we use to mark angles θ Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ Adjacent Hypotenuse The hypotenuse is always the same but the other 2 sides change if θ changes Opposite © T Madas
Hypotenuse Opposite θ Adjacent “thita” is a Greek letter we use to mark angles Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ Opposite Hypotenuse The hypotenuse is always the same but the other 2 sides change if θ changes θ Adjacent © T Madas
θ Hypotenuse Adjacent Opposite “thita” is a Greek letter we use to mark angles θ Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ Adjacent Hypotenuse The hypotenuse is always the same but the other 2 sides change if θ changes Opposite © T Madas
Hypotenuse Opposite θ Adjacent “thita” is a Greek letter we use to mark angles Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ Opposite Hypotenuse The hypotenuse is always the same but the other 2 sides change if θ changes θ Adjacent © T Madas
θ Hypotenuse Adjacent Opposite “thita” is a Greek letter we use to mark angles θ Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ Adjacent Hypotenuse The hypotenuse is always the same but the other 2 sides change if θ changes Opposite © T Madas
Hypotenuse Opposite θ Adjacent “thita” is a Greek letter we use to mark angles Opposite side: it lies opposite the angle θ Adjacent side: it touches the angle θ Opposite Hypotenuse The hypotenuse is always the same but the other 2 sides change if θ changes θ This will become important later Adjacent © T Madas
The Beginning of Trigonometry © T Madas
Opposite = = Adjacent = 8 Opposite θ Adjacent = 12 © T Madas
= = θ θ Opposite Adjacent = 4 Opposite Why is this angle also θ ? = 6 θ © T Madas
Opposite = = Adjacent = 6 Opposite θ Adjacent = 9 θ © T Madas
Opposite = = Adjacent = 5 Opposite θ Adjacent = 7.5 θ © T Madas
Opposite = Adjacent θ Opposite θ θ θ Adjacent © T Madas
θ Opposite θ θ θ Adjacent tangent of θ For a given acute angle of a right angled triangle: Opposite = constant tangent of θ Adjacent θ Opposite θ θ θ Adjacent © T Madas
θ Opposite θ θ θ Adjacent tangent of θ θ For a given acute angle of a right angled triangle: Opposite = tangent of θ θ Adjacent θ Opposite θ θ θ Adjacent © T Madas
Tangent Practice © T Madas
What is the tangent of θ ? What is the tangent of a ? 5 3 4 a θ Opposite Adjacent 3 4 tanθ = = = 0.75 Opposite Adjacent 4 3 tana = = ≈ 1.33 © T Madas
What is the tangent of x ? 13 5 12 x Opp Adj 12 5 tanx = = = 2.4 © T Madas
What is the tangent of y ? 17 15 8 Opp Adj 15 8 tany = = = 1.875 y © T Madas
What is the tangent of θ ? 25 7 24 θ Opp Adj 24 7 tanθ = = ≈ 2.43 © T Madas
In every right angled triangle: Opposite side = constant Adjacent side Opposite side = tanθ Adjacent side Every acute angle θ has its tanθ (constant ratio) stored in your calculator © T Madas
Out NOW! Calculators x2 x-1 x3 π . EXP Ans = 1 2 3 + – 4 5 6 x ÷ 7 8 9 . EXP Ans = 1 2 3 + – 4 5 6 x ÷ 7 8 9 DEL AC RCL ENG ( ) , M+ (–) . , ,, hyp sin cos tan a b/c x2 log ln x-1 nCr Pol( REPLAY ^ SHIFT ALPHA MODE ON x! nPr Rec( x3 d/c 10x ex sin-1 cos-1 tan-1 M- OFF STO π DRG› % Rnd Ran# A B C D E F X Y M ; e : CLR Calculators Out NOW! © T Madas
Find the tangent button in your calculator . EXP Ans = 1 2 3 + – 4 5 6 x ÷ 7 8 9 DEL AC RCL ENG ( ) , M+ (–) . , ,, hyp sin cos tan a b/c x2 log ln x-1 nCr Pol( REPLAY ^ SHIFT ALPHA MODE ON x! nPr Rec( x3 d/c 10x ex sin-1 cos-1 tan-1 M- OFF STO π DRG› % Rnd Ran# A B C D E F X Y M ; e : CLR Find the tangent button in your calculator © T Madas
tan 3 tan 3 = 0.577350269 x2 x-1 x3 π . EXP Ans = 1 2 3 + – 4 5 6 x ÷ . EXP Ans = 1 2 3 + – 4 5 6 x ÷ 7 8 9 DEL AC RCL ENG ( ) , M+ (–) . , ,, hyp sin cos tan a b/c x2 log ln x-1 nCr Pol( REPLAY ^ SHIFT ALPHA MODE ON x! nPr Rec( x3 d/c 10x ex sin-1 cos-1 tan-1 M- OFF STO π DRG› % Rnd Ran# A B C D E F X Y M ; e : CLR tan 3 tan 3 = 0.577350269 © T Madas
tan 6 4 tan 6 4 = 2.050303842 x2 x-1 x3 π . EXP Ans = 1 2 3 + – 4 5 6 . EXP Ans = 1 2 3 + – 4 5 6 x ÷ 7 8 9 DEL AC RCL ENG ( ) , M+ (–) . , ,, hyp sin cos tan a b/c x2 log ln x-1 nCr Pol( REPLAY ^ SHIFT ALPHA MODE ON x! nPr Rec( x3 d/c 10x ex sin-1 cos-1 tan-1 M- OFF STO π DRG› % Rnd Ran# A B C D E F X Y M ; e : CLR tan 6 4 tan 6 4 = 2.050303842 © T Madas
tan 2 9 tan 2 9 = 0.554309051 x2 x-1 x3 π . EXP Ans = 1 2 3 + – 4 5 6 . EXP Ans = 1 2 3 + – 4 5 6 x ÷ 7 8 9 DEL AC RCL ENG ( ) , M+ (–) . , ,, hyp sin cos tan a b/c x2 log ln x-1 nCr Pol( REPLAY ^ SHIFT ALPHA MODE ON x! nPr Rec( x3 d/c 10x ex sin-1 cos-1 tan-1 M- OFF STO π DRG› % Rnd Ran# A B C D E F X Y M ; e : CLR tan 2 9 tan 2 9 = 0.554309051 © T Madas
. EXP Ans = 1 2 3 + – 4 5 6 x ÷ 7 8 9 DEL AC RCL ENG ( ) , M+ (–) . , ,, hyp sin cos tan a b/c x2 log ln x-1 nCr Pol( REPLAY ^ SHIFT ALPHA MODE ON x! nPr Rec( x3 d/c 10x ex sin-1 cos-1 tan-1 M- OFF STO π DRG› % Rnd Ran# A B C D E F X Y M ; e : CLR tan-1 . 5 26.56505118 shift tan . 5 = 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
. EXP Ans = 1 2 3 + – 4 5 6 x ÷ 7 8 9 DEL AC RCL ENG ( ) , M+ (–) . , ,, hyp sin cos tan a b/c x2 log ln x-1 nCr Pol( REPLAY ^ SHIFT ALPHA MODE ON x! nPr Rec( x3 d/c 10x ex sin-1 cos-1 tan-1 M- OFF STO π DRG› % Rnd Ran# A B C D E F X Y M ; e : CLR tan-1 4 . 1 7 76.31552216 shift tan 4 . 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
. EXP Ans = 1 2 3 + – 4 5 6 x ÷ 7 8 9 DEL AC RCL ENG ( ) , M+ (–) . , ,, hyp sin cos tan a b/c x2 log ln x-1 nCr Pol( REPLAY ^ SHIFT ALPHA MODE ON x! nPr Rec( x3 d/c 10x ex sin-1 cos-1 tan-1 M- OFF STO π DRG› % Rnd Ran# A B C D E F X Y M ; e : CLR tan-1 . 5 3 3 28.05761573 shift tan . 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