Lesson 5.1 Using Fundamental Identities Essential Question: How do you rewrite trigonometric expressions to simplify and evaluate trigonometric functions?
Before we start… What is another way to write csc 𝜃 ?
Rewriting Trig Expressions Sometimes you will see a trigonometric expression that contains at least two different trig functions. We want to manipulate these using their reciprocal and ratio identities to try and rewrite using just one trig function.
Other times you want to use the Pythagorean identities to rewrite the trig expression.
How do I rewrite trigonometric expressions? To rewrite when there are no trig functions being squared: You want to rewrite each function in terms of sine and cosine. Once rewritten see if anything cancels. Rewrite as one trig function.
To rewrite if there are trig functions being squared: Use the Pythagorean identities to rewrite what you have been given. You may need to also rewrite using sine and cosine once you’ve used the Pythagorean identity. Rewrite as one trig function or as an addition fact with one trig function.
Using Fundamental Identities One common use of trigonometric identities is to use the given values of trigonometric functions to evaluate other trigonometric functions.
Use the values sec 𝑢 =− 3 2 and tan 𝑢 >0 to find the values of all six trigonometric functions.
Use the values sin 𝑢 = 1 2 and cos 𝑢 >0 to find the values of all six trigonometric functions.
Simplify sin 𝑥 cos 2 𝑥 − sin 𝑥
Simplify cos 2 𝑥 csc 𝑥 − csc 𝑥
Simplify sin 𝑡 + cot 𝑡 cos 𝑡
Simplify csc 𝑡− cos 𝑡 cot 𝑡
Simplify sec 𝑥 cos 𝑥
Simplify tan 𝑥 csc 𝑥
Simplify cot 2 𝜃 csc 2 𝜃
Simplify 1− cos 2 𝑥 csc 𝑥
Factor sec 2 𝜃−1
Factor 4 tan 2 𝜃+ tan 𝜃−3
Factor 1−cos 2 𝑥
Factor 2csc 2 𝑥−7 csc 𝑥 +6
Factor csc 2 𝑥− cot 𝑥 −3
Factor sec 2 𝑥+3 tan 𝑥 +1
Rewrite so that it’s not in fractional form. 1 1+ sin 𝑥
Rewrite so that it’s not in fractional form. cos 2 𝑦 1− sin 𝑦
How do you rewrite trigonometric expressions to simplify and evaluate trigonometric functions?
Ticket Out the Door Simplify sin 𝑥 tan 𝑥