Math-3 Lesson 6-1. Trigonometric Ratios for Right Triangles and Extending to Obtuse angles.

Transcription

1 Math-3 Lesson 6-1 Trigonometric Ratios for Right Triangles and Extending to Obtuse angles.

2 Right Triangle: has one angle whose measure is. 90 The short sides of the triangle are called legs. The side osite the right angle is called the hypotenuse. a leg c hypotenuse leg Standard Labeling: (for any triangle): Angles are labeled with capital letters. The sides osite each angle is labeled with the corresponding lower case letter. B Sides can also be named based upon their Endpoints. Side c is also segment AB. C b A

3 Triangle Similarity: If triangles are similar then there are 3 pairs of congruent angles and the ratio of corresponding sides (between the 2 triangles) will always equal the same number. A G B E C F AB XY BC YZ AC XZ

4 Do it right now! 1. Get into groups of 3 2. Sit with your group. 3. Count off in your groups (person 1, person 2, person 3).

5 Are these triangles Similar? You built these triangles in our activity cm 12 cm 14 cm

6 We defined similarity by making a ratio of a side of one triangle with the corresponding side of the other triangle. AB 5 1 CB In our activity we discovered that XY 10 2 YZ Ratios of sides of one triangle will always be equal to the Ratios of corresponding sides of a similar triangle. 4 8 AC BC 3 4 XZ YZ 6 8

7 Why is that so important? B 1 C A=30 Triangle Similarity is the result of corresponding congruent angles. X The ratios of sides will be a unique number for any given angle. A hyp ( for 30 angle) AB BC 1 2 XY YZ? Z Y=30 Y

8 Trigonometric Ratios of Right Triangles B a C Opposite leg b c Adjacent leg How many ratios of sides are possible for angle A? A Not all triangles have sides named a, b, c, so we generalize the location of the sides relative to one of the acute angles of the triangle. hypotenuse A hyp hyp a c c a adj hyp hyp adj b c c b adj adj hyp We use a code word for each ratio. a b b a

9 What are the code words for the ratios? The ratios are based upon their relative position to the angle. SOH-CAH-TOA sin cos tan hyp adj hyp adj SHA-CHO-CAO cosecant secant cotangent hyp hyp adj adj These only work for right triangles!!!

10 The ratio is a property of the angle. We must know the measure of the angle to find the numerical value of the ratio. You must: (1) State the code word for the ratio you want to use. sin (2) State which angle of the triangle to use. sin 30 The calculator (or table) will give the corresponding ratio. sin cos tan o h a h o a SOH-CAH-TOA csc sec cot h o h a adj SHA-CHO-CAO sin 30 h o 0.5 Angle: has units of either (1) radians or (2) degrees

11 Trigonometric ratios are not given in fraction form Angle hyp

12 0.174 C sine ratio sin hyp B X A=10 1 sin10 A Z D=10 Y

13 Sin ϴ= osite side Cos ϴ= adjacent side Tan ϴ= /adj hypotenuse = 1 Why is it nice to have a hypotenuse = 1? sine ratio hyp 1 The length of the hypotenuse is no longer in the ratio!

14 Code word Sin Sin Cos Cos Angle A B A B Ratio 3/ 5 4 / 5 4 / 5 3/ 5 B 5 3 C sin B cos A sin 30 cos60 sin A cos B sin 60 cos30 4 A Tan Tan Csc Csc Sec Sec Cot cot A B A B A B A B 5/ 3 5/ 3 3/ 4 4 / 3 5/ 4 5/ 4 4 / 3 3/ 4 Tan A (Tan B) Csc A (Sin A) Csc B (Sin B) Sec A Csc B Sec A (Cos A) Cot A Tan B Cot B (Cot A) 1 1 Tan 30 (Tan 60) Sec B Csc A 1

15 Trig Ratios of Acute Angles Go to Geogebra (trig ratios) What shape is used to define trig ratios? right triangle. What happens if the angle is greater than 90? Using these definitions we can t have angles > 90!!!

16 Trig ratios for obtuse angles: we need acute angles!! Sin ϴ= osite side Cos ϴ= adjacent side Tan ϴ= /adj

17 Standard Position Angle: 1) Angle is on the x-y plane 2) The vertex is at (0, 0) 3) Initial Side of the Angle is on x-axis pointing in the positive x direction. 4) terminal side of the angle points outward from the origin into any quadrant. y ϴ x

18 Standard Position Angle: If the terminal side is in quadrant 1, we can construct a nice Right triangle with hypotenuse = 1 y 1 Opp ϴ Adj x

19 Quadrant 1 Right Triangle: The (x, y) pair on the terminal side of the angle that is in quadrant 1 (a distance of 1 unit from the origin) is important. Length of the osite side of the angle = y-value of the point. y 1 (x, y) Opp Sin ϴ = Sin ϴ = y ϴ Adj x

20 Quadrant 1 Right Triangle: The (x, y) pair on the terminal side of the angle that is in quadrant 1 (a distance of 1 unit from the origin) is important. y 1 (x, y) Opp (adj, ) = (x, y) ϴ Adj Cos ϴ = adj. x Cos ϴ = x Length of the adjacent side of the angle = x-value of the point.

21 Standard Position Angle: If terminal side of the angle points outward from the origin into quadrant 2, 3, or 4, the angle is greater than 90 degrees. Sine, cosine, tangent ratios are defined based upon the sides of a right triangle! There is a very nice fix for this problem y ϴ x

22 Reference angle: The acute angle between the terminal side of a standard position angle and the x-axis (either (+) or (-) direction) y Opp ϴ Adj ϴ x We can construct a nice Right triangle with hyp = 1 using the reference angle instead of the measure of the standard position angle.

23 The (x, y) pair on the terminal side of the angle that is in any quadrant (a distance of 1 unit from the origin) is important. Length of the osite side of the angle = y-value of the point. (adj, ) = (x, y) (x, y) 1 Opp y ϴ Sin ϴ = Sin ϴ = y ϴ Adj x Length of the adjacent side of the angle = x-value of the point. Cos ϴ = adj. Cos ϴ = x

24 We can find the sine, cosine, tangent ratios for any angle. We just have to keep track of the sign (+/-) of the ratio. The sign (+/-) of the ratio comes from the sign (+/-) of the (x, y) pair according to its quadrant. y Opp Adj ϴ-180 ϴ x (adj, ) = (x, y)

25 y ϴ Adj ϴ x 1 Opp

26 The (x, y) pair of the point on the terminal side of the angle is exactly 1 unit distance from the origin. This point will lie on a circle around the origin whose radius = 1. We call this circle the Unit Circle y x, adj, cos, sin r = 1 Ɵ adj Geogebra Unit Circle