Simplifying trigonometric ratios. Grade 10 and 11 Trigonometry

1. TRIGONOMETRY
SIMPLIFYING TRIGONOMETRIC EXPRESSIONS

2. AIMS AND OBJECTIVES
• The aim of the lesson:
• To teach learners how to simplify trigonometric expressions
using the CAST diagram.
• To teach learners how to simplify trigonometric expressions
using identities and Co-ratios

3. PRIOR KNOWLEDGE
• In grade you learnt about the theorem of Pythagoras, the
trigonometric ratios and the names of the sides on a right
angle.
• Recap Questions
1. State the theorem of Pythagoras
2. Define sin x, cos x and tan x in terms of their sides
3. Name the sides of the triangle.

4. PRIOR KNOWLEDGE
Solve for x using the theorem of Pythagoras.
Revision
Theorem of Pythagoras: 𝑥2
+ 𝑦2
= 𝑟2
, where c is the hypotenuse.
𝑆𝑖𝑛𝑥 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
=
𝑦
𝑟

5. PRIOR KNOWLEDGE
• 𝐶𝑜𝑠𝑥 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
=
𝑥
𝑟
• 𝑡𝑎𝑛𝑥 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
=
𝑦
𝑥

6. THE CAST DIAGRAM
• This CAST diagram helps us to identify whether our
trigonometric ratios are positive or negative.

7. CAST DIAGRAM AND TRIGONOMETRIC
RATIOS.
• Use the CAST diagram to write the trigonometric ratios in terms of x
• Sin (180°+x)= -Sin x
• Cos (360°-x)= Cos x
• Tan (90°+x)= -tan x
Co-ratios
90° is a special angle that has a unique effect to Cos and Sin.

8. CO-RATIOS
• 𝑆𝑖𝑛 (90°−x) =Cos x
• 𝐶𝑜𝑠 90° + 𝑥 = − sin 𝑥
This effect is applicable with the angle 270° as well.
What will be the answer:
𝑆𝑖𝑛 (90°+x) = ??
𝑆𝑖𝑛 (270° − 𝑥)= ??

9. TRIGONOMETRIC IDENTITIES
𝑡𝑎𝑛𝑥 =
𝑠𝑖𝑛𝑥
𝑐𝑜𝑠𝑥
𝑠𝑖𝑛2
𝑥 + 𝑐𝑜𝑠2
𝑥 = 1
Therefore, 𝑠𝑖𝑛2
𝑥= 1- 𝑐𝑜𝑠2
𝑥
And 𝑐𝑜𝑠2 𝑥= 1- 𝑠𝑖𝑛2 𝑥

10. SIMPLIFYING A TRIGONOMETRIC
EXPRESSION

11. CLASSWORK
1. In which Quadrants are the following positive:
• Sin θ
• Cos θ
• Tan θ
• 2. Express the following in terms of θ:
• Sin (90°+ θ)
• Tan (180°- θ)
• Cos (360°- θ)

12. CLASSWORK
• 3. Simplify the expressions
• 𝒔𝒊𝒏 𝟗𝟎° − x cos 360° − x tan 180° − x ÷ cos(180° − x)
The end.