Primary Ratios ~ Soh Cah Toa: tan θ = O/A tan θ = 7/4 θ = tan -1 (7/4) θ = 60 o

Primary Ratios ~ SohCah Toa: tan θ = O/A tan θ = 7/4 θ = tan-1(7/4) θ = 60o Secondary Ratios ~ Inverse of Primary Ratios: Sine = Cosecant Cosine = Secant Tangent = Cotangent 7 eg. θ 4

Remember: • Sine ratios are the opposite of Cosine • Ratios may be rationalized to eliminate root from the denominator • eg.

(FIRST DAY) Remember: • There are 4 quadrants • Q1 begins where both the X and Y value are positive • It then continues in a counter clockwise motion

Terminology • Initial arm:Angle arm lying on x-axis • Terminal Arm:Angle arm that rotates (moves). • When the initial arm is on the positive x-axis (whose vertex is at the origin), the position is called standard position • The angle created is referred to as θ(measured from the initial arm to the terminal arm) • Principal Angle: counter-clockwise angle between the initial arm and the terminal arm. (this is angle θ) • Related Acute Angle:The angle between the terminal arm and the x-axis, when the terminal arm lies in Quadrants II, III or IV. (this is angle β)

CAST System • Used to help you remember which ratio is positive in which quadrant (letter represents the +’ve ratio) • C (IV) – Cosine + • A (I) – All + • S (II) – Sine + • T (III) – Tangent + • Equations • C = (360°- θ) • A = not available • S = (180° - θ) • T = (180°+ θ)

Steps to Finding the Related Acute Angle • Example 1 • sin 210° (principal angle is 210°) • After drawing angle, determine • which quadrant terminal angle will fall in • Then use CAST system to conclude whether sine is + or - in that quadrant (Quadrant III, -) • Determine the related angle α (angle between terminal arm and x-axis - 30°) • Write the corresponding exact value/ratio -

(SECOND DAY) The Unit Circle • Radius is always 1 unit • QuadrantalAngle: Lies on the axis (i.e. 90°, 180°, 270°, 360°) • Recall • The 30-60-90 triangle: The radius is 2, this will not fit into the unit circle •  The entire triangle must be divided by 2: y = x = r = 1 y =1 x =√3 r = 2  2

The same goes for the 60-30-90 triangle (where x and y switch values) • The 45-45-90 triangle, √2 won’t fit into the unit circle: •  The entire triangle must be divided by √2 x = y = r = 1 x = 1 y = 1 r = √2  √2

The Unit Circle sin θ = cosθ= tan θ=

Any point (x,y) can determine an angle • Example: • Point (4,5) is on the terminal angle. Find the 3 primary trigonometric ratios and the corresponding angle β • With β, the corresponding angle θcan also be found using the CAST formulas • First determine the radius by using the Pythagorean Theorem • In this case r = 6.4 or 41 (keep exact values) • SIN   Corresponding angle (β) is (Sin-1) 51° • TAN   Corresponding angle(β) is 51° • COS   Corresponding angle(β) is 51° • ** The principal angle can be determined by using the CASTequations **

What are they? • An equation involving trigonometric ratios that are true no matter which value you substitute for the variable • For angles, we use Theta (θ) or • Alpha (α)

What do we need to remember? • There are many formulas we need to memorize in order to be successful at solving Identities. (Reciprocal ones as well)

Things to look for when solving identities • Always look for patterns • Try to manipulate the formula to create one of the formulas on the previous side, so that you can simplify your equation. • Always use LS/RS! You will lose marks if you don’t • Look for ways to factor your sides • Finding a difference of squares, solving an equation that looks like a parabola, etc. • Find common denominators whenever possible; it will make the simplifying process easier

Use When Given The Following: • ASS • ASA • AAS

Ambiguous Case of Sine Law • If you are given two sides and one angle (where youmust find an angle): • The Sine Law could possibly provide youwith one or more solutions (even none) So ... how do you get the value 112.9°? 180° - 67.1° = 112.9° Remember! • The sum of the interior angles is 180° • Triangles can’t have two obtuse angles

When do we use it? Formula: • As indicated in the formula, if we’re given ALL THREE SIDES, we can solve for cosC, and then use inverse to find out the angle (SSS) • If we’re given TWO SIDESand theANGLE BETWEEN THEM, we can plug in the values to determine the remaining side (SAS)

Application Questions Bearing– A clockwise angle from magnetic north Lighthouse is located at abearingis 335° -x- 360° – 335° = 25°

Example #1 • Radar stations A and B are on an east-west line, 3.7 km apart. Station A detects a plane at C, on a bearing of 61°, station B detects the plane at a bearing of 321°. Find the distance from A to C. 61°

61° • We can use Sine Law to solve side AC (b) • HINT: Use “Z” patterns

You can solve 3D problems using combinations of: • Trig ratios • Pythagorean Theorem • The Sine Law • The Cosine Law Depending on given info Easy way to remember The 3 axis

Example #1 (trig ratios) Determine value of ‘x’ tan E = tan 35° = e = 15 * tan 35° e = 10.5 cm ( is also “g” ) sin G = sin 45° = sin 45° = 10.5 = x sin 45° x = x = 14.85 x = 15 cm g e

Primary Ratios ~ Soh Cah Toa: tan θ = O/A tan θ = 7/4 θ = tan -1 (7/4) θ = 60 o