Trigonometry Review August 14-15, 2017

Why do we need trigonometry? Trig allows us to calculate the sides or angles of right triangles We will use trig constantly in the first three quarters of physics … basically anytime something happens at an angle. Examples: Finding resultant velocity of a plane that travels first in one direction, then another Calculating the time, path, or velocity of a ball thrown at an angle Predicting the course of a ball after a collision Calculating the strength of attraction between charges in space etc., etc., etc

Right triangles The formulas that we learn today work only with right triangles … but that’s ok, we can create a right triangle to solve any physics problem involving angles! But, it does beg the question … what’s a right triangle? a triangle with a 90o angle

Calculating the length of the sides of a right triangle If you know the length of two of the sides, then use … the Pythagorean Theorem: c2 = a2 + b2 Example: A = 3 cm, B = 4 cm, what is C? C2 = (3cm)2 + (4cm) 2 C2 = 25cm2 C = 5 cm NOTE: “C” always refers to the hypotenuse! The hypotenuse is always the longest side and its always the side that is opposite of the right angle.

Work on individually. Find the missing side. z 9 m 1 mm y 2 mm x 6 cm 2 cm 7 m Z = 2 mm X = 6 m Y = 6 cm

Calculating the length of the sides of a right triangle What if we have one side and one angle? How do we find the other sides? We can use the trig functions: sin, cos, and tan sin θ = Opposite / Hypotenuse cos θ = Adjacent / Hypotenuse tan θ = Opposite / Adjacent

Examples: y 25 cm θ = 25 degrees 5 m x θ = 30 degrees Find y and x

25 cm 5 m x Find y and x sin(30) = y/25cm tan (25) = 5m / x Examples: y 25 cm θ = 25 degrees 5 m x θ = 30 degrees Find y and x sin(30) = y/25cm tan (25) = 5m / x 25cm*sin(30) = y x = 5m / tan(25) 13 cm = y x = 11 m

Examples: θ = 35 degrees z y 18 cm 6 m θ = 40 degrees Find z and y

z y 18 cm 6 m Find z and y sin (40) = 18 cm / z tan (35) = y / 6m Examples: θ = 35 degrees z y 18 cm 6 m θ = 40 degrees Find z and y sin (40) = 18 cm / z tan (35) = y / 6m z = 28 cm y = 4 m

Calculating the angles of right triangle In any triangle (right or not) the angles add to 180o. Example: Find a A = 180 – 70 – 50 = 60o

Calculating the angles of right triangle In right triangles, we can also find the angle using the side lengths and inverse trig functions sin-1 (opp / hyp) = θ cos-1 (adj / hyp) = θ tan-1 (opp / adj) = θ

Examples: φ = ? 25 cm 6 m 18 cm 11 m θ = ? Find θ and φ

sin-1 (18cm / 25cm) = θ tan-1 (6 m / 11m) = φ Examples: φ = ? 25 cm 6 m 18 cm 11 m θ = ? Find θ and φ sin-1 (18cm / 25cm) = θ tan-1 (6 m / 11m) = φ θ = 46 degrees φ = 29 degrees

Examples: φ = ? θ = ? 12 m 8cm 9cm 10 m Find θ and φ

tan-1 (8cm / 9cm) = θ cos-1 (10 m / 12m) = φ Examples: φ = ? θ = ? 12 m 8cm 9cm 10 m Find θ and φ tan-1 (8cm / 9cm) = θ cos-1 (10 m / 12m) = φ θ = 42 degrees φ = 34 degrees

Mixed Practice Find all sides and angles

Closure, HW, & Exit Ticket What were our objectives today, and how well did we accomplish them? How did we address our unit statement today? What was our LP trait and how did we demonstrate it? HW – Trig HW (HW QUIZ NEXT CLASS) START VECTORS IF POSSIBLE!