28 December 2012TrigonometryLearning Objective:To be able to describe the sides of right-angled triangle for use in trigonometry.Trigonometry is concerned with theconnection between the sides andangles in any right angled triangle. Angle
The sides of a right -angled triangle aregiven special names:The hypotenuse, the opposite and theadjacent.The hypotenuse is the longest side and isalways opposite the right angle.The opposite and adjacent sides refer toanother angle, other than the 90o. A A
There are three formulae involved intrigonometry: sin A= cos A= tan A = S OH C AH T OA
Using trigonometry on the calculator 28 December 2012Learning Objective:To be able to use a scientific calculator tofind decimal values and angles intrigonometry.All individual angles have different sine, cosineand tangent ratios (or decimal values).Scientific calculators store information aboutevery angle.We need to be able to access thisinformation in the correct manner.
Finding the ratiosThe simplest form of question is finding thedecimal value of the ratio of a given angle. Find: 1) sin 32 = sin 32 = 2) cos 23 = 3) tan 78 = 4) tan 27 = 5) sin 68 =
Using ratios to find anglesWe have just found that a scientificcalculator holds the ratio informationfor sine (sin), cosine (cos) andtangent (tan) for all angles.It can also be used in reverse, findingan angle from a ratio.To do this we use the sin-1, cos-1 andtan-1 function keys.
Example:1. sin x = 0.1115 find angle x. sin-1 0.1115 = ( shift sin ) x = sin-1 (0.1115) x = 6.4o2. cos x = 0.8988 find angle x cos-1 0.8988 = ( shift cos ) x = cos-1 (0.8988) x = 26o
28 December 2012TrigonometryLearning Objective:To be able to use trigonometry to find theunknown angle in a triangle.
Finding an angle from a triangleTo find a missing angle from a right-angledtriangle we need to know two of the sides ofthe triangle.We can then choose the appropriate ratio,sin, cos or tan and use the calculator toidentify the angle from the decimal value ofthe ratio.1. Find angle C a) Identify/label the 14 cm names of the sides. b) Choose the ratio that C contains BOTH of the 6 cm letters.
1. We have been given H the adjacent and 14 cm hypotenuse so we use COSINE: C adjacent Cos A = 6 cm hypotenuse A Cos A = a h Cos C = 6 14 Cos C = 0.4286 C = cos-1 (0.4286) C = 64.6o
2. Find angle x Given adj and opp x need to use tan:3 cm A opposite Tan A = adjacent 8 cm O Tan A = o a Tan x = 8 3 Tan x = 2.6667 x = tan-1 (2.6667) x = 69.4o
3. Given opp and hyp 12 cm need to use sin:10 cm opposite y Sin A = hypotenuse sin A = o h sin x = 10 12 sin x = 0.8333 x = sin-1 (0.8333) x = 56.4o
28 DecemberTrigonometryLearning Objective:To be able to use trigonometry to find anunknown side in a triangle.
Finding a side from a triangleTo find a missing side from a right-angledtriangle we need to know one angle and oneother side. Note: If x Cos45 = 13 To leave x on its own we need to move the ÷ 13. It becomes a “times” when it moves. Cos45 x 13 = x
1. We have been given H the adj and hyp so we 7 cm use COSINE: adjacent Cos A = hypotenuse 30o k A Cos A = a h Cos 30 = k 7 Cos 30 x 7 = k 6.1 cm = k
2. We have been given the opp and adj so we 50o use TAN:4 cm A Tan A = r O Tan A = o a Tan 50 = r 4 Tan 50 x 4 = r 4.8 cm = r
3. We have been given H the opp and hyp so wek 12 cm use SINE:O Sin A = 25o sin A = o h sin 25 = k 12 Sin 25 x 12 = k 5.1 cm = k
Finding a side from a triangleThere are occasions when the unknownletter is on the bottom of the fraction aftersubstituting. Cos45 = 13 u Move the u term to the other side. It becomes a “times” when it moves. Cos45 x u = 13 To leave u on its own, move the cos 45 to other side, it becomes a divide. u = 13 Cos 45
28 DecemberTrigonometryLearning Objective:To be able to use trigonometry to find anunknown side when the unknown letter ison the bottom of the fraction.When the unknown letter is on the bottomof the fraction we can simply swap it withthe trig (sin A, cos A, or tan A) value. Cos45 = 13 u u = 13 Cos 45
1. Cos A = a h H x Cos 30 = 5 x 5 30 o x = cos 30 5 cm A x = 5.8 cm sin A = o 2. h H8 cm m sin 25 = 8 m O m= 8 25o sin25 m = 18.9 cm
28 DecemberTrigonometryLearning Objective:To be able to use trigonometry to findunknown sides and unknown angles in atriangle.
1. Cos A = a h H x Cos 30 = 5 x 5 30o x = cos 30 5 cm A x = 5.8 cm2. Tan A = o a 50o Tan 50 =r4 cm 4 A Tan 50 x 4 = r r 4.8 cm = r O 3. o sin A = h 12 cm sin y = 10 10 cm 12 y sin y = 0.8333 y = sin-1 (0.8333) y = 56.4o