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Examples of Trigonometric Relationships

Having looked at the various trigonometric relationships, we now look at some examples of trigonometric relationships

 

Example 1: sin 40° =  cos x°. What is x?

Sine and Cosine are complementary ratios, hence

x  =  90° – 40°  =  50°

Therefore sin 40°  =  cos 50°, and

x  =  50°

 

Example 2: cos 60° = sin x°. What is x?

x  =  90° – 60° =  30°

Therefore cos 60°  =  sin 30°, and

x  =  30°

 

Example 3: Simplify {cos 70°}/{sin 20°}

{cos 70°}/{sin 20°}~~=~~{cos (90°~-~20°)}/{sin 20°}~~=~~{sin 20°}/{sin 20°} =  1

 

Example 4: Find the value of θ if tan θ  =  {cos 40°}/{cos 50°}

{cos 40°}/{cos 50°}~~=~~{cos 40°}/{cos(90°~-~40°)}~~=~~{cos 40°}/{sin 40°} = cot 40°

tan θ  =  {cos 40°}/{cos 50°}  =  cot 40°  =  tan (90 – 40°)

tan θ  =  tan (90 – 40°)  =  tan 50°

Hence θ  =  50°

 

Example 5:  Find the value of θ if cot θ  =  {cos 40°}/{cos 50°}

cot θ  =  {cos 40°}/{cos 50°}  =  {cos 40°}/{cos(90°~-~40°)}~~=~~{cos 40°}/{sin 40°}  =  cot 40°

Hence θ  =  40°

 

Example 6: If cos θ  =  1/2, what is sec θ?

sec θ  =  1/{cos theta}~~=~~1/(1/2)  =  2

 

examples of trigonometric relationships-1Example 7:  If cos θ  =  1/2, what is cosec θ?

Using Pythagoras theorem, x  =  sqrt {2^2~-~1^2}~~=~~sqrt {3}

sin θ  =  {sqrt 3}/2.

Hence  cosec θ  =  1/{sin theta}~~=~~2/{sqrt 3}