Bell Ringer ( 5 mins in notebook) Find his horizontal velocity? X

Bell ringer Find his horizontal velocity?

Lesson Objectives We will analyze motion in 2 Dimensions using vectors & trigonometry I will use trigonometry to analyze motion in 2 Dimensions by finding the direction (sides) and the angle. DQ: How do I find 1 side when I have 2 sides? How do I solve for the missing angle when I have 2 sides ?How do I solve for 1 missing side when I have the angle and 1 other side?

AGENDA Review- Pythagorean theorem & Triangles Intro to Trig functions Worksheet: Using calculator Worksheet: Solving with trig Physics Application problems

Makiah &Christiana- October 17th!!!

Callie Handy’s October 19th!!!

BUT……what about the direction? since it is a VECTOR we should include a DIRECTION on our final answer. N W of N E of N N of E N of W W E N of E S of W S of E NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE. W of S E of S S

BUT…..what about the VALUE of the angle??? Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VALUE of the angle. To find the value of the angle we use a Trig function called TANGENT. 109.8 km 55 km, N q N of E 95 km,E So the COMPLETE final answer is : 109.8 km, 30 degrees North of East

Trig functions Represent the ratio of 2 sides of a triangle Has special names Helps find a missing side or angle of a right triangle

Trigonometry Obj: I can to use trigonometry to find unknown sides and unknown angles in a triangle. Trigonometry is concerned with the connection between the sides and angles in any right angled triangle. Angle

The sides of a right -angled triangle are given special names: The hypotenuse, the opposite and the adjacent. The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides refer to another angle, other than the 90o. A

The Trigonometric Functions we will be looking at SINE COSINE TANGENT

The Trigonometric Functions SINE COSINE TANGENT

SINE Prounounced “sign”

Prounounced “co-sign” COSINE Prounounced “co-sign”

Prounounced “tan-gent”

Represents an unknown angle Greek Letter q Prounounced “theta” Represents an unknown angle

hypotenuse hypotenuse opposite opposite adjacent adjacent

We need a way to remember all of these ratios…

Some Old Hippie Came A Hoppin’ Through Our Old Hippie Apartment

Sin SOHCAHTOA Opp Hyp Cos Adj Hyp Tan Opp Adj Old Hippie

Using trigonometry on the calculator All individual angles have different sine, cosine and tangent ratios (or decimal values). Scientific calculators store information about every angle. We need to be able to access this information in the correct manner.

Finding the ratios The simplest form of question is finding the decimal value of the ratio of a given angle. Find: sin 32 = sin 32 = 0.5514

Finding a side from a triangle To find a missing side from a right-angled triangle we need to know one angle and one other side. Note: If Cos45 = To leave x on its own we need to move the ÷ 13. It becomes a “times” when it moves. Cos45 x 13 = x

Finding a side from a triangle There are occasions when the unknown letter is on the bottom of the fraction after substituting. Cos45 = 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 =

When the unknown letter is on the bottom of the fraction we can simply swap it with the trig (sin A, cos A, or tan A) value. Cos45 = u =

7 cm k 30o 4. We have been given the adj and hyp so we use COSINE: Cos A = H A Cos A = Cos 30 = Cos 30 x 7 = k 6.1 cm = k

4 cm r 50o 5. We have been given the opp and adj so we use TAN: Tan A = A O Tan A = Tan 50 = Tan 50 x 4 = r 4.8 cm = r

12 cm k 25o 6. We have been given the opp and hyp so we use SINE: Sin A = H O sin A = sin 25 = Sin 25 x 12 = k 5.1 cm = k

Practice

x 5 cm 30o 7. Cos A = H Cos 30 = x = A x = 5.8 cm sin A = m 8 cm 25o 8. H sin 25 = O m = m = 18.9 cm

sin 30 = 0.5 cos 30 = 0.866 tan 30 = 0.5774 sin 50 = 0.766 cos 50 = 0.6428 tan 50 = 0.1.1917

Using ratios to find angles We have just found that a scientific calculator holds the ratio information for sine (sin), cosine (cos) and tangent (tan) for all angles. It can also be used in reverse, finding an angle from a ratio. To do this we use the sin-1, cos-1 and tan-1 function keys.

( ) ( ) Example: sin x = 0.1115 find angle x. sin-1 0.1115 = shift sin x = 6.4o 2. cos x = 0.8988 find angle x cos-1 0.8988 = shift cos ( ) x = cos-1 (0.8988) x = 26o

Finding an angle from a triangle To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle. We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio. 14 cm 6 cm C 1. Find angle C Identify/label the names of the sides. b) Choose the ratio that contains BOTH of the letters.

14 cm 6 cm C 1. We have been given the adjacent and hypotenuse so we use COSINE: Cos A = H A Cos A = Cos C = Cos C = 0.4286 C = cos-1 (0.4286) C = 64.6o

Find angle x 2. 8 cm 3 cm x Given adj and opp need to use tan: Tan A = A O Tan A = Tan x = Tan x = 2.6667 x = tan-1 (2.6667) x = 69.4o

3. 12 cm 10 cm y Given opp and hyp need to use sin: Sin A = sin A = sin x = sin x = 0.8333 x = sin-1 (0.8333) x = 56.4o

Summary To find 1 side when I have 2 sides of a right triangle and I will use________ ________. To solve for the missing angle when I have 2 sides I can use the _________ trig function. To solve for 1 missing side when I have the angle and 1 other side I can use_________.

Summary To find 1 side when I have 2 sides of a right triangle and I will use________ ________. To solve for the missing angle when I have 2 sides I can use the _________ trig function. To solve for 1 missing side when I have the angle and 1 other side I can use_________. Pythagorean theorem

Summary To find 1 side when I have 2 sides of a right triangle and I will use________ ________. To solve for the missing angle when I have 2 sides I can use the _________ trig function. To solve for 1 missing side when I have the angle and 1 other side I can use_________. Pythagorean theorem Inverse

Summary To find 1 side when I have 2 sides of a right triangle and I will use________ ________. To solve for the missing angle when I have 2 sides I can use the _________ trig function. To solve for 1 missing side when I have the angle and 1 other side I can use_________. Pythagorean theorem Inverse trig functions ( SOH CAH TOA)