Use of Sine, Cosine and Tangent TRIGONOMETRY Use of Sine, Cosine and Tangent

SIMILAR TRIANGLE REVISION Similar triangles have corresponding angles that are equal The sides in one triangle are in a given ratio to the sides in the other triangle. The larger triangles are scaled up versions of the small triangle. RATIO = 2 1.5 cm RATIO = 3 1cm Ө 3 cm 1.12 cm 2 cm 4.5 cm 3 cm Ө 2.24 cm Ө 3.36 cm

SIMILAR TRIANGLE REVISION Similar triangles have corresponding angles equal The sides in one triangle are in a given ratio to the sides in the other triangle. The larger triangles are scaled up versions of the small triangle.

SIMILAR TRIANGLE REVISION The larger triangles are scaled up versions of the small triangle.

TRIGONOMETRY some definitions The side opposite the angle Ө is the Opposite side (Opp) The side opposite the Right angle is the Hypotenuse (Hyp) The side next to the angle is the Adjacent (Adj) Hypotenuse Opposite Ө Adjacent

TRIGONOMETRY Hypotenuse Opposite Ө Adjacent

TRIGONOMETRY Hypotenuse Opposite Ө Adjacent

TRIGONOMETRY Remember: The sides in one triangle are in a given ratio to the sides in the other triangle. BUT: The sides within the one triangle are also related to each other in ratios which depend on the angle Ө. RATIO = 2 1.5 cm RATIO = 3 1cm Ө 3 cm 1.12 cm 2 cm 4.5 cm 3 cm Ө 2.24 cm Ө 3.36 cm

TRIGONOMETRY - Sine The sides within the one (right angled) triangle are related to each other in ratios which depend on the angle Ө. Opp = 1 = 2 = 3 = Sine Ө = Hyp 1.5 3.0 4.5 0.67 1.5 cm 1cm Ө 3 cm 1.12 cm 2 cm 4.5 cm 3 cm Ө 2.24 cm Ө 3.36 cm

TRIGONOMETRY - Cosine The sides within the one (right angled) triangle are related to each other in fixed ratios which depend on the angle Ө. Adj = 1.12 = 2.24 = 3.36 = Cosine Ө = Hyp 1.5 3.0 4.5 0.74 1.5 cm 1cm Ө 3 cm 1.12 cm 2 cm 4.5 cm 3 cm Ө 2.24 cm Ө 3.36 cm

TRIGONOMETRY - Sine For right angled triangles, these ratios have been recorded and are stored in your calculator Opp = 2 = 0.67 = Sine Ө = Sin Ө = Sin42˚ Hyp 3 Angle Ө = 42 degrees 3 cm 2 cm 2.24 cm

TRIGONOMETRY - Cosine Adj = 2.24 = 0.74 = Cosine Ө = Cos Ө = Cos42˚ Hyp 3 Angle Ө = 42 degrees 3 cm 2 cm 2.24 cm

TRIGONOMETRY - Tangent Opp = 2 = 0.89 = Tangent Ө = Tan Ө = Tan42˚ Adj 2.24 Angle Ө = 42 degrees 3 cm 2 cm 2.24 cm

Can you guess what X could be? TRIGONOMETRY Any right angled triangle with an angle of 42 ˚ will have sides in the same ratio as these triangles. Can you guess what X could be? Angle Ө = 42 degrees X metres (Opposite) 100 metres (Adjacent)

TRIGONOMETRY Tan42˚ = Opp = X Adj 100 . . 0.89 = X 100 . . X = 89 metres Angle Ө = 42 degrees X metres (Opposite) 100 metres (Adjacent)

Can you guess what Y could be? TRIGONOMETRY Can you guess what Y could be? 89 m Angle Ө = 42 degrees Y metres

TRIGONOMETRY Can you guess what Y could be? Tan42˚ = Opp = 89 Adj Y . . . Tan42˚ = 89 = 0.89 Y . . Y = 1 89 0.89 . . Y = 1 x 89 0.89 = 100 m 89 m Ө Y metres Angle Ө = 42 degrees

TRIGONOMETRY - applications A four metre length of string, attached to a bolt at the top of a brick wall, makes an angle of 48 degrees with the ground What is the height (B) of the brick wall? Ө 4 m B = ?

TRIGONOMETRY Draw a diagram Which information do you have? Sin = Opp/Hyp Cos = Adj/Hyp Tan = Opp/Adj TRIGONOMETRY Draw a diagram Which information do you have? Which information do you want? Which function do you use? Write an equation Solve the equation Angle and Hypotenuse Opposite side, B Sine Sin 48 = B/4 Ө 4 m B = ? If B = Sin 48 = 0.74 4 B = 0.74 . . . B = 0.74 X 4 . . B = 2.96 m

TRIGONOMETRY

TRIGONOMETRY