TRIGONOMETRIC ANGLES : in four quadrants

Describe transformation of sin(x-10) into sin(x+10)

- 20 units in total

Solve equation for interval -360 <x<360

- ALL VALUES INCLUDING POSITIVR AND NEGATIVR - consider the interval first

AREA OF TRIANGLE OBSERVATION

- ALWAYS WRITE 2 possible value sof the angle when we must use sin graph - LOOK AT THE CONTEXT OF HE TRIANGLE AND DECIIDE WHICH VALUE OF X SEEMS MORE APPROPRIATE

If the transformation is x+n eg sin(x+60) it is...

- ALWYAS TRANSLATE TO THE LEFT BY 60 units - MAX AND MINIMUM STAY THE SAME

MAJOR TIP BEFORE ANSWERING TRIANGLE OR TRIG QUESTIONS

- Analyse the provided diagram - to see which angles are OBTUSE AND WHICHA ARE ACUTE

Cos Graph INTERVALS

- Cosx= cos (360-x) - Cosx = cos(360+x) - Cos(-x) = -cos(x)

WHAT NOT TO DO IN HARD TRIG EQUATIONS

- DRAW THE TRANSFORMED graph - UNLESS IT TELLS YOU IN THE QUESTION YOU MUST SKETCH ONE - Not necessary and time consuming - in reality you are drawing a graph for u

What is helpful in solving x or determining the number of solutions?

- DRAWING A GRAPH : even if you think you know all the answers

Given tan x = 1/2 is obtuse/acute/reflex find sin or cosx

- FIRST TO TAN INVERSE TO FIND THE ANGLE - Then sub in that angle in both sin and cosx - form the identity jf required

Identify 2 mistakes made by student

- IN SIMPLYFYING THE EQUATION THEY FOUND ADDTIONA SOLUTION AFTER DIVIDING BY THE factor of angle - NOT APPLIED FOR FULL INTERVl of solutions ( not provided all correct values)

Identify error made by student A

- It could be something as simple as rearringing - this is why you should TRY TO SOLVE YOURSELF USING CORRECT METHOD

How to find k in tankx = 1/2 if x=60 is a solution

- Let kx = u - Solve u by doing inverse tan - then let that = kx - Which is infact 60k - Solve k finally

Function can be modelled using y=sin(30t) where t is the time If t=0 represents midday find the least possible times for at lest half fjll using the

- Let u =30t and solve the sin functjon to get u = 30 degrees and 180-30 =150 degrees - So t = 1 and t=5 - Between 1 and 5 pm

State WITH JUSTIFICATION if this is the only possible value of k

- NI - Increasing K will bring another branch if the tan graph into place

CAST DIAGRAM

- Since word is spelled out from bottom eight going anticlockwise

Sine Graph INTERVALS

- Sinx = sin(180-x) - Sinx = (360+x) - Sin(-x)= -sin(x)

Tips : when asked to describe the transformatijjn

- State in what direction the tranformation is done if it si stretch or transatjoj - Eg stretch x values by 3 in y. Direction

Even if angles lie outside the range (360)

- Still eb found in one of the four quadrants - 600-360 = 240 - so it would like in the third quadrant

Where are Exact Values derived from ?

- The RIGHT ANGLES TRIANGLES

Identify erro made by student B and explain effect on their answer

- They have SQUARED THE SIMPLE EQUATION - But SQUARING CAN Lead to MORE SOLUTIONS

SKETCH FUNCTION FOR GIVEN INTERVAL

- This means dont plot degrees along the x axis but use he UNITS OF THE INTERVAL GIVEN - Then estimate the shape of the graph in terms of what should be incldued and what shouldbt - Eg the questjons specifies an interval we may not need to do for any negative values -

HARD TRIANGLE PROBLEMS

- Tip : if there is a lack in informationtry to FORM ALGEBRAIC EXPRESSION and EQUATIONS - then we would have to solve x

Eliminate e the angles to form an equation of x and y

- USE THE 2 TRIGONOMETRIC IDENITIES GET = 1 - Then replace with x and y

IMPORTANT TUIP🍈

- USE YOUR CALCULATOR TO DOUVLE CHECK THE X VALUE YOU HAVE CALCULATED IS CORRECT

Given sin x / cos x only = k/l find the exact value of cos x and sin x

- Use IDENTITIES - Sin2 x + cos 2 x =1 - Sin x / cos x = tan x - LEAVE IN EXACT FORM T

Proof this is a right angles isosceles triangle

- Using the area formula: - But as θ is not the largest angle, θ must be 45. Use the cosine rule to find x.(. As we have to image only one of the side penths is given and we are trying to find unknown side length - So the triangle is isosceles with two angles of 45°. It is a right-angled isosceles triangle

UPDATED : Given tanx = k/l find the value of cosx and sinx

- We first use a UNIT CIRCLE TO DRAW THE TRIANGLE GIVEN IT either acute / obtuse or reflex - this helps us determine if cos(x) / sin(x) are negative or positive - Form a TRIANGLE with the x axis - Cos x = x vales so the BASE OF THE triangle - SIn x = y vales si the height of the trusnglr - look at the fraction and place the top number as he height and the denominator as cos - DO pythagoras to find the hypotenuse - Use SOH CAH TOA to find the exact value of cos x and sin x

Tip for PROOF USING IDENTITIES

- break down an expression of sine or cos or tan ss simply down as possible - At end make a clear comment showing how it equals to RHS or LHS

Tip : DOUBLE CHECK IF ANSWER IS RIGHT

- by using calcualtor to see if the exact values match up

When angle is obtuse?

- cos (o) is negative - Sin (o) is positive - Tan (o) is negative

When angle is acute?

- cos (o) is positive - Sin (o) is positive - Tan (o) is positive

Tips for sketching horrible trig grpahs

- do 2 units on graph = 45 degrees along the x axis - 4 units =1 -

MAIN TIPS HARDER TRIG TRIANGLES

- if question is LONGWINDED PLAN SEQUENCE IF METHOD - ALWAYS RETAIN TOUR VALUES IN YOUR CALCULATOR WHEN DOING MULTIPLE CALCULATIONS - TRY TO CONSIDER THE CONDITONS OF TRIG TRIANGLES MET( eg 2 slide lengths and angle not enclosed is amibgous sine rule) - TRY TO SEPERSTE SHAPE IF POSSIBLE

Colinear points questikn

- may invovle angles of elevation and depression - require us to find the HEIGH OF SOMETHING AT THE 3 rs point - Asymptoon is all angles measured from ground level

SQUARE ROOT AND SOLVING EQUATIONS

- must take into consideration all possible balues - so do NEGATIVE AND POSITIV ROOTS

Evidence he graph is periodic

- point P correpsonding to an angle - Is same as point P corresponding to an angle + 360 - Shows graphs y= sin (x) and y= cos(x) are periodic with 360 degrees

Tan Graph INTERVALS

- tanx = tan(180+ x) - Tan(-x) = -tan x

What I NEED TO START TO APPRECIATE :

- there can be more than one value for a solutjon ESPECIALLY WITH SIN - always dihble check how many answers the question wants - And also IF THEY ARE WITHIN THE CORRECT INTERVAL

HENCE FIND EXACT VALUE OF COS OR SIN X AFTER USING COSINE OR SINE RULE

- use the TRIGONOMETRIC IDENTITIES - only worth 2-1 mark

Tip for cos interval

- whether it is TOP OR BOTTOM PEAKS AND TROUGHS OF THE GRAPH TRY TO alswyas do 360-x

What direction to do we go to work out positive angle ?

Anticlockwise

If the transformation is (70-x) then you

Change to (x-70)

What is a unit circle?

Circle with radius of 1 unit

What direction do we go to work out negative angle?

Clockwise

Interval in sketching trig graohs

DO NOT CHANGE ONLY SKETCH FOR THE INTERVAL GIVEN

Tip for BEARINGS INVOLVING COS AND SINE RULE

Extend your line to north and south line

⇒tan−θ= 1 Look at the graph of y = tanθ in the interval 0 ≤ θ ≤ 360° . There are two solutions.

NOTE IN THIS INTERVAL WE ARE NOT CONSIDERING NEGATIVE SIDR

Third Quadrant

Ranged up to 270 degrees

Second quadrant (anti-clockwise)

Ranges up to 180 degrees

Fourth Quadrant

Ranges up to 360 degrees

First quadrant

Ranges up to 90 degrees

Intervals for sin

Sinx= sin(180-x) Sinx= sin(360+x)

HOW TO DETERMINE THE PERIOD OF THE GRPAH

Sketch it and look for where it repeats

EXTREMELY IMPORTANT HOW TO EXPRESS ANSWER

TRIGONOMETRIC RATIO

Suggest why not a realistic model

The sand dunes may not all be same height

Where do we draw the triangle from?

The x axis no matter what quadrant the angle lands in

Write down one assumption to the model you have made

Theassumptionisthattheballswings symmetrically.

From the graph drawn suggest the number of sand dunes

There are 4 complete waves so there are four complete sand dunes

Tip factorising hard expressions

Try ti arrange all similar terms together like circle questions

To determine where graph crosses

Where x=0 SUB IN X = 0 then SOLVE THE WHAT THE FUNCTJON IS EQUAL TO

To determine where graph crosses the x acis

Where y=o Do the inverse functjon if nessessary of read kf the graph by looking at trasnformation

If the transformation is x-n eg sin(x-60) it is

—. ALWAYS TRANSLATE TO THE RIGHT BY 60 unjta - Vector ( 60 0)