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In this lesson, we will learn how to use the Pythagorean identity and double-angle formulas to evaluate trigonometric values.

Q1:

Find the value of c o s 2 π΄ given c o s π΄ = β 3 5 where 9 0 < π΄ < 1 8 0 β β without using a calculator.

Q2:

Find the value of c o s οΌ π΄ 2 ο given s i n οΌ π΄ 2 ο = 3 5 without using a calculator.

Q3:

Find, without using a calculator, the value of s i n 2 π΄ given t a n π΄ = β 5 1 2 where 3 π 2 < π΄ < 2 π .

Q4:

Find the value of c o s ο½ π 2 ο given c o s π = 1 5 1 7 where 0 < π < 9 0 β β without using a calculator.

Q5:

Knowing that 5 π₯ + 1 2 π₯ = 1 3 s i n c o s , find s i n π₯ and c o s π₯ .

Q6:

Knowing that 3 π₯ β 4 π₯ = 5 s i n c o s , find s i n π₯ and c o s π₯ .

Q7:

Find, without using a calculator, the value of s i n 2 π΄ given c o s π΄ = β 1 2 1 3 where 1 8 0 β€ π΄ < 2 7 0 β β .

Q8:

Find, without using a calculator, the value of s i n ο½ π 2 ο given t a n π = β 1 5 8 where 3 π 2 < π < 2 π .

Q9:

π΄ π΅ πΆ is a triangle where t a n πΆ = 8 1 5 . Find the value of s i n οΌ π΄ + π΅ 2 ο .

Q10:

Find the value of c o s ( π + 2 π΄ ) given s i n ( 2 7 0 + π΄ ) = β 1 5 1 7 β where 3 π 2 < π΄ < 2 π .

Q11:

Find, without using a calculator, 1 β 2 π 1 + 2 π c o s c o s given t a n π = 4 where π β ο π , 3 π 2 ο .

Q12:

Which of the following is equal to β 1 β 2 π₯ s i n ?

Q13:

Use the addition formula to find an expression for s i n 2 πΌ .

Q14:

Given that s i n c o s π + π = β 7 1 3 and π < π < 3 π 2 , determine the possible values of c o s 2 π .

Q15:

Find the value of 1 + 2 π΄ 1 + 2 π΄ s i n c o s given t a n π΄ = 5 2 6 where 0 < π΄ < π 3 without using a calculator.

Q16:

Find the value of t a n c o t 1 5 7 3 0 β² + 1 5 7 3 0 β² β β and then t a n c o t 2 β 2 β 1 5 7 3 0 β² + 1 5 7 3 0 β² without using a calculator.

Q17:

Find the value of s i n 4 π given 2 π π β 2 π π = 9 2 6 s i n c o s c o s s i n 3 3 .

Q18:

Find, without using a calculator, the value of t a n 4 π given s i n c o s π π = β 1 4 where π β ο π 2 , 3 π 4 ο .

Q19:

Use the addition formula to find an expression for c o s 2 πΌ .

Q20:

Which of the following is equal to β 1 β 2 π₯ c o s ?

Q21:

Using the half angle formulas, or otherwise, find the exact value of t a n ο» π 8 ο .

Q22:

Find the value of 1 β ο» ο 1 + ο» ο t a n t a n 2 7 π 8 2 7 π 8 without using a calculator.

Q23:

Evaluate 3 6 1 1 2 3 0 β² 1 β 1 1 2 3 0 β² t a n t a n β 2 β without using a calculator.

Q24:

Simplify t a n t a n 3 5 4 4 β² 1 4 β² β² 1 β 3 5 4 4 β² 1 4 β² β² β 2 β .

Q25:

π΄ π΅ πΆ is a triangle, where the ratio between its lengths π , π , and π is 4 βΆ 3 βΆ 5 . Find t a n 2 π΄ .

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