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# UPREP Trig Math Workbook Grade 11

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```Grade 11
TRIGONOMETRY
here
23 July 2022
TRIGONOMETRY
NB - make sure your calculator is on ‘DEG’
1. DEFINITIONS, IDENTITIES &amp; RULES:
1.1 Definitions:
𝑌
For any size 𝜃 :
𝑦
⚫
sin 𝜃 =
⚫
cos 𝜃 = 𝑟
⚫
tan 𝜃 = 𝑥
𝑟
𝑥
𝑦
(𝑥 ; 𝑦)
=
⚫
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑟
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝜃
𝑋
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
=
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
=
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
1.2 Identities:
sin 𝜃
1
tan 𝜃 = cos 𝜃
⚫
⚫
sin2 𝜃 + cos2 𝜃 = 1
⚫
sin2 𝜃 = 1 − cos2 𝜃
⚫
cos2 𝜃 = 1 − sin2 𝜃
tan 𝜃
=
cos 𝜃
⚫
sin 𝜃
1.3 180&deg; Rule:
90&deg;
2) Use CAST diagram to determine the sign
3) Keep the same ratio (sin, cos or tan), but
possibly change the sign.
e.g. sin(−180&deg; + 𝜃 ) = − sin 𝜃
because (−180&deg; + 𝜃) is in the 3rd
180&deg; − 𝜃
180&deg;
𝜃
𝑆
𝐴
𝑇
𝐶
180&deg; + 𝜃
0&deg;
360&deg;
360&deg; − 𝜃
270&deg;
1.4 90&deg; Rule:
⚫
⚫
sin(90&deg; − 𝜃 ) = cos 𝜃
cos(90&deg; − 𝜃 ) = sin 𝜃
⚫
⚫
+
sin(90&deg; + 𝜃 ) = cos 𝜃
cos(90&deg; + 𝜃 ) = −sin 𝜃
−
2. SPECIAL ANGLES:
2.1 Special triangles:
Know the 45&deg;/30&deg;/60&deg; ratios without a
calculator.
Use calculator for 0&deg;/90&deg;/180&deg;/270&deg;/360&deg;
⚫ sin 45&deg; =
1
√2
⚫ cos 45&deg; =
1
√2
√2
1
45&deg;
1
30&deg;
⚫ tan 45&deg; = 1
2
⚫ sin 60&deg; =
√3
2
⚫ cos 60 &deg; =
1
2
⚫ tan 60&deg; = √3
⚫ sin 30&deg; =
1
2
√3
⚫ cos 30 &deg; =
√3
2
⚫ tan 30&deg; =
1
√3
60&deg;
1
2.2 Combined with 180&deg; and 90&deg; rules:
e.g. cos 150&deg; = − cos 30&deg;
ALWAYS first reduce to an acute angle.
= −
√3
2
3. USING SKETCHES:
Y
3.1 If 5 cos 𝜃 = −4 , use a sketch (and not a
calculator) to find sin 𝜃 if
180&deg; ≤ 𝜃 ≤ 360&deg;
⚫
⚫
⚫
𝜃
X
Must be quadrant 3 where cos is negative.
Use Pythagoras to calculate 𝑦.
Watch out for the sign.
3.2 If tan 20&deg; =
𝑝
3
⚫
⚫
3
𝑦 = −3 , therefore sin 𝜃 = − 5
, write sin2 20&deg; in terms
√𝑝 2 + 9
of 𝑝 .
⚫
5
(−4 ; 𝑦) ⚫
𝑝
20&deg;
Draw a right-angled triangle and fill in
𝑝 and 3 on the opposite and adjacent
sides.
3
rd
⚫
Use Pythagoras to calculate the 3 side.
2
sin 20&deg; = (
𝑝
√𝑝2 +9
2
) =
𝑝2
𝑝2 +9
4. TRIG EQUATIONS:
4.1
⚫
Equations must be manipulated to give:
⚫
Watch out for −1 ≤ the number ≤ 1 for
sin / cos 𝜃
⚫
First find the help-angle:
⚫
Then use CAST-diagram to determine
⚫
sin / cos / tan 𝜃 = a number
⚫
SHIFT-button (don’t push minus)
sin / cos : + 360&deg;𝑛 ; 𝑛 ∈ ℤ
tan : + 180&deg;𝑛 ; 𝑛 ∈ ℤ
Find all angles in interval by substituting
𝑛 = 1 , −1 , 2 , etc
4.2
⚫
General solution:
⚫
⚫
⚫
Interval given:
⚫
4.3
⚫
sin 𝛼 = 3 cos 𝛼
⚫
Divide by cos 𝛼 to get tan 𝛼
⚫
sin 𝛼 = cos 𝛽
⚫
Use the 90&deg; rule to change sin ↔ cos
4.4 Factorising:
eg.
eg.
eg.
eg.
sin 𝑥 + 3 sin 𝑥 cos 𝑥 = 0
sin2 𝑥 − 9cos2 𝑥 = 0
sin2 𝑥 − 2 sin 𝑥 cos 𝑥 + cos2 𝑥 = 0
6 cos 𝑥 − 3 cos2 𝑥 + 2 sin 𝑥 cos 𝑥 − 4 sin 𝑥 = 0
⚫
⚫
⚫
⚫
⚫
Common factor
Difference of squares
Trinomial
Grouping
Sum &amp; Difference of cubes
TRIGONOMETRIE
NB – maak seker jou sakrekenaar is op ‘DEG’
1. DEFINISIES, IDENTITEITE &amp; RE&Euml;LS:
1.1 Definisies:
𝑌
Vir enige grootte 𝜃 :
𝑦
⚫
sin 𝜃 =
⚫
cos 𝜃 = 𝑟
⚫
tan 𝜃 =
𝑟
𝑥
𝑦
𝑥
=
(𝑥 ; 𝑦)
⚫
𝑜𝑜𝑟𝑠𝑡𝑎𝑎𝑛𝑑𝑒
𝑟
𝑠𝑘𝑢𝑖𝑛𝑠𝑠𝑦
𝜃
𝑋
𝑎𝑎𝑛𝑔𝑟𝑒𝑛𝑠𝑒𝑛𝑑
=
𝑠𝑘𝑢𝑖𝑛𝑠𝑠𝑦
𝑜𝑜𝑟𝑠𝑡𝑎𝑎𝑛𝑑𝑒
=
𝑎𝑎𝑛𝑔𝑟𝑒𝑛𝑠𝑒𝑛𝑑
1.2 Identiteite:
sin 𝜃
1
cos 𝜃
⚫
tan 𝜃 = cos 𝜃
⚫
⚫
sin2 𝜃 + cos2 𝜃 = 1
⚫
sin2 𝜃 = 1 − cos2 𝜃
⚫
cos2 𝜃 = 1 − sin2 𝜃
tan 𝜃
=
sin 𝜃
1.3 180&deg; re&euml;l:
90&deg;
2) Gebruik CAST-diagram om teken te
bepaal
3) Behou dieselfde verhouding (sin, cos of
tan), maar verander dalk die teken.
bv. sin(−180&deg; + 𝜃 ) = − sin 𝜃
want (−180&deg; + 𝜃) is in die 3de
180&deg; − 𝜃
180&deg;
𝜃
𝑆
𝐴
𝑇
𝐶
180&deg; + 𝜃
0&deg;
360&deg;
360&deg; − 𝜃
270&deg;
1.4 90&deg; re&euml;l:
⚫
⚫
sin(90&deg; − 𝜃 ) = cos 𝜃
cos(90&deg; − 𝜃 ) = sin 𝜃
⚫
⚫
+
sin(90&deg; + 𝜃 ) = cos 𝜃
cos(90&deg; + 𝜃 ) = −sin 𝜃
−
2. SPESIALE HOEKE:
2.1 Spesiale driehoeke:
Ken die 45&deg;/30&deg;/60&deg; verhoudings sonder ‘n
sakrekenaar.
Gebruik ‘n sakrekenaar vir
0&deg;/90&deg;/180&deg;/270&deg;/360&deg;
⚫ sin 45&deg; =
1
√2
⚫ cos 45&deg; =
1
√2
√2
1
45&deg;
1
30&deg;
2
⚫ tan 45&deg; = 1
√3
⚫ sin 60&deg; =
√3
2
⚫ cos 60 &deg; =
1
2
⚫ tan 60&deg; = √3
⚫ sin 30&deg; =
1
2
⚫ cos 30 &deg; =
√3
2
⚫ tan 30&deg; =
1
√3
60&deg;
1
2.2 Met 180&deg; en 90&deg; re&euml;ls gekombineer:
bv. cos 150&deg; = − cos 30&deg;
Maak ALTYD eerste ‘n skerphoek.
= −
√3
2
3. GEBRUIK VAN SKETSE:
Y
3.1 As 5 cos 𝜃 = −4 , gebruik ‘n skets (en nie
‘n sakrekenaar) om sin 𝜃 te vind as
180&deg; ≤ 𝜃 ≤ 360&deg;
⚫
⚫
⚫
𝜃
X
Moet kwadrant 3 wees waar cos negatief is.
Gebruik Pythagoras om 𝑦 te bereken.
Oppas vir die teken.
(−4 ; 𝑦) ⚫
⚫
5
3
𝑦 = −3 , dus is sin 𝜃 = − 5
𝑝
3.2 As tan 20&deg; = 3 , skryf sin2 20&deg; in terme
van 𝑝 .
⚫
⚫
√𝑝 2 + 9
Teken ‘n reghoekige driehoek en vul
𝑝 en 3 op die teenoorstaande en
aangrensende sye in.
𝑝
20&deg;
3
Gebruik Pythagoras om die 3de sy te vind.
⚫
2
sin 20&deg; = (
𝑝
√𝑝2 +9
2
) =
𝑝2
𝑝2 +9
4. TRIGONOMETRIESE VERGELYKINGS:
4.1
⚫
Vergelykings moet herrangskik word as:
⚫
Oppas vir −1 ≤ die getal ≤ 1 vir
sin / cos 𝜃
⚫
Vind eerstens die hulphoek:
⚫
Gebruik dan die CAST-diagram om die
⚫
sin / cos / tan 𝜃 = ‘n getal
⚫
SHIFT-knoppie (moenie minus druk nie)
4.2
⚫
Algemene oplossing:
⚫
⚫
sin / cos ∶ + 360&deg;𝑛 ; 𝑛 ∈ ℤ
tan : + 180&deg;𝑛 ; 𝑛 ∈ ℤ
⚫
Interval gegee:
⚫
Vind alle hoeke in interval deur substitusie
𝑛 = 1 , −1 , 2 , ens.
4.3
⚫
sin 𝛼 = 3 cos 𝛼
⚫
Deel deur cos 𝛼 om tan 𝛼 te kry
⚫
sin 𝛼 = cos 𝛽
⚫
Gebruik die 90&deg; re&euml;l vir sin ↔ cos
4.4 Faktorisering:
bv.
bv.
bv.
bv.
sin 𝑥 + 3 sin 𝑥 cos 𝑥 = 0
sin2 𝑥 − 9cos2 𝑥 = 0
sin2 𝑥 − 2 sin 𝑥 cos 𝑥 + cos2 𝑥 = 0
6 cos 𝑥 − 3 cos2 𝑥 + 2 sin 𝑥 cos 𝑥 − 4 sin 𝑥 = 0
⚫
⚫
⚫
⚫
⚫
Gemene faktor
Drieterm
Groepering
Verskil en som van derdemagte
EXERCISE: Mixed Trig questions
1. If cos 160&deg; =
1
, determine, without a calculator,
𝑝
the value of sin 250&deg; in terms of p.
(4)
2. Prove the following identities:
1
2
1
1+cos 𝛼
2.1 (sin 𝛼 + tan 𝛼) = 1−cos 𝛼
2.2
1−cos 𝑥
sin 𝑥
=
sin 𝑥
(5)
(4)
1+cos 𝑥
3. If 1 − sin 𝐴 = 𝑝 and 1 + 𝑠𝑖𝑛 𝐴 = 𝑞,
determine cos 2 𝐴 in terms of 𝑝 and 𝑞.
4. Solve : sin2 (𝜃 − 50&deg;) =
1
4
(3)
; 0&deg; ≤ 𝜃 ≤ 360&deg;
(6)
5. If sin 𝐴 = 𝑝 and cos 𝐵 = −√1 − 𝑝2 , where 𝑝 &gt; 0 ,
90&deg; &lt; 𝐴 &lt; 360&deg; and 180&deg; &lt; 𝐵 &lt; 360&deg; , prove that
𝐴 + 𝐵 = 360&deg;.
(5)
6. If
1
sin 𝜃
=
1
2𝑘
+
𝑘
2
and 90&deg; ≤ 𝜃 ≤ 270&deg; , determine,
without a calculator, the value of 𝑐𝑜𝑠𝜃 in terms of 𝑘 ,
𝑘 &gt; 1.
(6)
7. Simplify without a calculator :
√4 sin 150&deg; &times; 2 3 tan 225&deg;
(4)
8. Evaluate without a calculator:
sin 237&deg; cos 147&deg; −
cos 303&deg; cos 213&deg;
tan 237&deg;
9. If 𝛼 + 𝜃 = 90&deg; and tan 𝛼 =
1
𝑘
(6)
;𝑘≠0,
simplify in terms of 𝑘 :
cos(𝜃−720&deg;)
sin2 (180&deg;+𝛼).cos(90&deg;−𝜃)
10.1 Prove the identity :
cos 𝑥
1+sin 𝑥
(8)
+
1+sin 𝑥
cos 𝑥
=
2
cos 𝑥
(5)
10.2 For which values of 𝑥 in the interval
0&deg; ≤ 𝑥 ≤ 360&deg; will the identity be undefined?
(3)
11. Calculate the value of 𝐴 if
𝐴 = 2sin(𝑎−𝑚)&deg; . sin(𝑏−𝑚)&deg; . sin(𝑐−𝑚)&deg;.
(3)
12. Determine the general solution :
2 sin 𝑥 tan 𝑥 + tan 𝑥 = 2 sin 𝑥 + 1
. . . . sin(𝑧−𝑚)&deg;
(7)
13. Determine, without a calculator, the value of :
𝑠𝑖𝑛2 1&deg; + 𝑠𝑖𝑛2 2&deg; + 𝑠𝑖𝑛2 3&deg;+ . . . +𝑠𝑖𝑛2 87&deg; + 𝑠𝑖𝑛2 88&deg; + 𝑠𝑖𝑛2 89&deg;
(4)
14. Simplify without a calculator :
√3 sin 𝑥 sin2 72&deg; + sin2 198&deg; √3 cos(𝑥−90&deg;)
tan 120&deg; sin 𝑥
(6)
15. Determine the general solution :
√tan 𝜃 = 𝑥 +
1
if
𝑥
16. Determine the value of
𝑎 if cos 𝜃 =
17. If sin 𝜃 =
calculate
5
𝑎−1
𝑎+1
1
𝑥 2 + 𝑥2 = 3
1
tan2 𝜃
(5)
1
+ sin2 𝜃 in terms of
; 𝑎 ≠ −1 and 0&deg; &lt; 𝜃 &lt; 90&deg;
, 90&deg; &lt; 𝜃 &lt; 270&deg; , 𝜃 = 90&deg; + 𝛼 ,
13
cos 𝛼
tan 𝜃
without a calculator.
(6)
(7)
18. Given : √5 cos 𝜃 − 2 = 0
Determine, without a calculator, the values
of 𝑎 and 𝑏.
(7)
𝑌
𝑃(𝑏; 4)
•
• 𝑀(6; 𝑎)
𝜃
𝑂
𝑋
19. 𝑃 and 𝑀 are points on a circle with centre 𝑂.
𝑅 is a point on the 𝑥 axis. If 𝑃𝑂̂𝑀 = 115&deg;,
calculate the values of 𝑎 and 𝑏.
(6)
𝑌
𝑂
• 𝑃(4; 3)
𝑅
•
• 𝑀(𝑎; 𝑏)
𝑋
20. In the diagram, 𝑃𝑂̂𝑄 = 90&deg;.
Determine the coordinates of 𝑄.
(7)
𝑌
• 𝑃(𝑥; √3)
2
𝜃
𝛼
𝑂
𝑋
20
• 𝑄(𝑎; 𝑏)
21. 𝑂𝑃 = 2𝑅𝑂 and sin 𝛼 =
3
5
Determine, without a calculator,
the coordinates of 𝑃.
(6)
𝑌
• 𝑅(𝑥; 3)
𝛽
𝛼
𝑂
•𝑃
[138]
•
𝑇
𝑋
```