1. CS1550 Fundamentals For Computer Graphics Trigonometry
Sumanth Shankar
California State University, Los Angeles

2. Trigonometric Relationships
The intimate relationship between sin and cos is cos (𝛽) = sin (𝛽+90º) Pythagoras theorem is used to derive following relationships 1) tan (𝛽) = sin (𝛽) cos (𝛽)

3. Trigonometric Relationships
2) 𝑠𝑖𝑛 2 𝛽 + 𝑐𝑜𝑠 2 𝛽 =1 3) 1+𝑡𝑎𝑛 2 𝛽 = 𝑠𝑒𝑐 2 𝛽 4) 1+𝑐𝑜𝑡 2 𝛽 = 𝑐𝑜𝑠𝑒𝑐 2 𝛽

4. Sine rule
The sine rule relates angles and side lengths for a triangle. Consider an arbitrary triangle C b a A B c

5. Sine rule
The sine rule states 𝑎 sin 𝐴 = 𝑏 sin 𝐵 = 𝑐 s𝑖𝑛 𝐶

6. The cosine rule The cosine rules are 𝑎 2 = 𝑏 2 + 𝑐 2 −2𝑏𝑐 𝑐𝑜𝑠 𝐴
𝑎 2 = 𝑏 2 + 𝑐 2 −2𝑏𝑐 𝑐𝑜𝑠 𝐴
𝑏 2 = 𝑐 2 + 𝑎 2 −2𝑐𝑎 𝑐𝑜𝑠 𝐵
𝑐 2 = 𝑎 2 + 𝑏 2 −2𝑎𝑏 𝑐𝑜𝑠 𝐶

7. The cosine rule
4) a =𝑏 𝑐𝑜𝑠 𝐶 +𝑐 cos 𝐵 5) b =𝑐 𝑐𝑜𝑠 𝐴 +𝑎 cos 𝐶 6) c =𝑎 𝑐𝑜𝑠 𝐵 +𝑏 cos 𝐴

8. Compound Angles
Two sets of compound trigonometric relationships show how to add and subtract two different angles and multiples of the same angle.
sin(A±𝐵)= sin 𝐴 cos 𝐵 ± cos(A)sin(B)
cos(A±𝐵)= cos 𝐴 cos 𝐵 ±sin(A)sin(B)

9. Compound Angles
3) tan(A±𝐵) = tan 𝐴 ± tan⁡(𝐵) 1± tan 𝐴 tan⁡(𝐵)

10. Perimeter Relationships
The perimeter of a triangle is calculated using s = 1 2 (a + b + c) The relationships that integrate angles with the perimeter of a triangle are: 1) sin 𝐴 2 = √ (𝑠−𝑏)(𝑠−𝑐) 𝑏𝑐

11. Perimeter Relationships
2) cos 𝐴 2 = √ 𝑠(𝑠−𝑎) 𝑏𝑐 3) sin(A) = 2 𝑏𝑐 √𝑠(𝑠−𝑎)(𝑠−𝑏)(𝑠−𝑐)