1. 2.1 – Trigonometric Functions of Acute Angles
Math 150
2.1 – Trigonometric Functions of Acute Angles

2. Another way you can define the trig functions is directly from right triangles. sin 𝜃 = opp hyp csc 𝜃 = hyp opp cos 𝜃 = adj hyp sec 𝜃 = hyp adj tan 𝜃 = opp adj cot 𝜃 = adj opp

3. Another way you can define the trig functions is directly from right triangles. sin 𝜃 = opp hyp csc 𝜃 = hyp opp cos 𝜃 = adj hyp sec 𝜃 = hyp adj tan 𝜃 = opp adj cot 𝜃 = adj opp
SOH CAH TOA

5. Ex 1. Find the sine, cosine, and tangent values for angles 𝐴 and 𝐵 in the following right triangle.

6. Since 𝐴 and 𝐵 are complementary angles and sin 𝐴 = cos 𝐵 , sine and cosine are called cofunctions. Also, 𝐴+𝐵= 90 ∘ , so 𝐵= 90 ∘ −𝐴, thus sin 𝐴 = cos 90 ∘ −𝐴 . This is one of the cofunction identities.

7. Since 𝐴 and 𝐵 are complementary angles and sin 𝐴 = cos 𝐵 , sine and cosine are called cofunctions. Also, 𝐴+𝐵= 90 ∘ , so 𝐵= 90 ∘ −𝐴, thus sin 𝐴 = cos 90 ∘ −𝐴 . This is one of the cofunction identities.

8. Since 𝐴 and 𝐵 are complementary angles and sin 𝐴 = cos 𝐵 , sine and cosine are called cofunctions. Also, 𝐴+𝐵= 90 ∘ , so 𝐵= 90 ∘ −𝐴, thus sin 𝐴 = cos 90 ∘ −𝐴 . This is one of the cofunction identities.

9. Since 𝐴 and 𝐵 are complementary angles and sin 𝐴 = cos 𝐵 , sine and cosine are called cofunctions. Also, 𝐴+𝐵= 90 ∘ , so 𝐵= 90 ∘ −𝐴, thus sin 𝐴 = cos 90 ∘ −𝐴 . This is one of the cofunction identities.

10. Since 𝐴 and 𝐵 are complementary angles and sin 𝐴 = cos 𝐵 , sine and cosine are called cofunctions. Also, 𝐴+𝐵= 90 ∘ , so 𝐵= 90 ∘ −𝐴, thus sin 𝐴 = cos 90 ∘ −𝐴 . This is one of the cofunction identities.

11. Cofunction Identities
sin 𝐴 = cos 90 ∘ −𝐴
sec 𝐴 = csc 90 ∘ −𝐴
tan 𝐴 = cot 90 ∘ −𝐴
cos 𝐴 = sin 90 ∘ −𝐴
csc 𝐴 = sec 90 ∘ −𝐴
cot 𝐴 = tan 90 ∘ −𝐴

12. Ex 2. Find one solution for the following equation
Ex 2. Find one solution for the following equation. Assume all angles involved are acute angles. cos (𝜃+ 4 ∘ ) = sin (3𝜃+ 2 ∘ )

13. There are two “special” triangles that can help us evaluate trig functions of 30 ∘ , 45 ∘ , and 60 ∘ angles.

14. Ex 3. Find the six exact trig function values for a 60 ∘ angle.

15. Note: The word “exact” means not approximated
Note: The word “exact” means not approximated. For example, is exact, whereas (a decimal approximation of ) is not exact because it was rounded to the nearest thousandths place. As a general rule, always give exact answers unless otherwise told.