Lesson: Angle Sum and Difference Identities

In this lesson, we will learn how to derive the angle sum and difference identities, graphically or using the unitary circle, and use them to find trigonometric values.

Sample Question Videos

  • 03:03

Worksheet: Angle Sum and Difference Identities • 25 Questions • 1 Video

Q1:

Given that s i n c o s c o s s i n s i n 6 0 3 0 6 0 3 0 = 𝜃 , find the value of 𝜃 in degrees.

Q2:

Given that c o s 𝜃 = 5 7 , where 3 𝜋 2 𝜃 2 𝜋 , and c o s 𝜑 = 2 3 , where 0 𝜑 𝜋 2 , find the exact value of c o s ( 𝜑 𝜃 ) .

Q3:

Simplify t a n t a n t a n t a n ( 1 1 8 2 𝑋 ) + ( 3 2 + 2 𝑋 ) 1 ( 1 1 8 2 𝑋 ) ( 3 2 + 2 𝑋 ) .

Q4:

Evaluate t a n t a n t a n t a n 1 + 5 𝜋 6 2 𝜋 3 5 𝜋 6 2 𝜋 3 .

Q5:

Evaluate t a n c o t c o t t a n 1 6 + 7 6 1 7 6 1 6 .

Q6:

Find c s c ( 𝐴 + 𝐵 ) given s i n 𝐴 = 3 5 where 2 7 0 < 𝐴 < 3 6 0 and c o s 𝐵 = 2 4 2 5 where 1 8 0 < 𝐵 < 2 7 0 .

Q7:

In triangle 𝐴 𝐵 𝐶 , 𝐴 and 𝐵 are acute angles, where s i n 𝐴 = 4 5 and c o s 𝐵 = 3 5 . Without using a calculator, find the value of c o s 𝐶 .

Q8:

Let us consider the two shown figures, each showing two points on the unit circle.

How is the figure on the right obtained from the figure on the left?

What can you say about the triangles 𝑂 𝑀 𝑁 and 𝑂 𝑀 𝑁 ?

Find the coordinates of 𝑀 , 𝑁 , 𝑀 , and 𝑁 .

Calculate the lengths of 𝑀 𝑁 and 𝑀 𝑁 .

Use what you found in the previous questions to find an expression for c o s ( 𝛼 𝛽 ) .

Q9:

Find s i n ( 𝐴 𝐵 ) given s i n 𝐴 = 5 1 3 where 2 7 0 < 𝐴 < 3 6 0 and c o s 𝐵 = 4 5 where 0 < 𝐵 < 9 0 .

Q10:

𝐴 𝐵 𝐶 is a triangle where c o s 𝐴 = 3 5 and s i n 𝐵 = 4 5 . Find s i n 𝐶 without using a calculator.

Q11:

Find s i n ( 𝐴 𝐵 ) given s i n 𝐴 = 4 5 where 9 0 < 𝐴 < 1 8 0 and c o s 𝐵 = 3 5 where 1 8 0 < 𝐵 < 2 7 0 .

Q12:

Find s i n ( 𝐴 + 𝐵 ) given c o s 𝐴 = 2 4 2 5 and t a n 𝐵 = 1 2 5 where 𝐴 and 𝐵 are two positive acute angles.

Q13:

Find c s c ( 𝐴 + 𝐵 ) given s i n 𝐴 = 4 5 where 9 0 < 𝐴 < 1 8 0 and c o s 𝐵 = 5 1 3 where 1 8 0 < 𝐵 < 2 7 0 .

Q14:

Using the relation s i n s i n c o s c o s s i n ( 𝛼 𝛽 ) = 𝛼 𝛽 𝛼 𝛽 , find an expression for s i n ( 𝛼 + 𝛽 ) .

Q15:

Find s i n ( 𝐴 𝐵 ) given s i n 𝐴 = 4 5 where 2 7 0 < 𝐴 < 3 6 0 and c o s 𝐵 = 4 5 where 1 8 0 < 𝐵 < 2 7 0 .

Q16:

Find c s c ( 𝐴 𝐵 ) given c o s 𝐴 = 7 2 5 and c o s 𝐵 = 3 5 where 𝐴 and 𝐵 are acute angles.

Q17:

Given that t a n 𝜃 = 3 4 , where 𝜃 is a positive acute angle, determine s i n ( 𝜃 + 6 0 ) without using a calculator.

Q18:

Which of the following is equivalent to 2 1 + 3 ?

Q19:

Given that t a n 𝐴 = 2 4 7 , where 0 < 𝐴 < 9 0 , and t a n 𝐵 = 8 1 5 , where 9 0 < 𝐵 < 1 8 0 , determine s i n ( 𝐴 𝐵 ) .

Q20:

Find s i n ( 𝐴 + 𝐵 ) given c o s 𝐴 = 4 5 and c o s 𝐵 = 3 5 where 𝐴 and 𝐵 are acute angles.

Q21:

Given that s i n 𝐴 = 3 5 and c o s 𝐵 = 1 2 1 3 , where 𝐴 and 𝐵 are two positive acute angles, determine s i n ( 𝐴 + 𝐵 ) .

Q22:

Given that t a n 𝐴 = 4 3 and t a n 𝐵 = 7 2 4 , where 𝐴 and 𝐵 are two positive acute angles, determine s i n ( 𝐴 + 𝐵 ) .

Q23:

Find s i n ( 𝑋 𝑌 ) given 2 5 𝑋 + 7 = 0 c o s where 9 0 < 𝑋 < 1 8 0 and c o s 𝑌 = 4 5 where 2 7 0 < 𝑌 < 3 6 0 .

Q24:

If s i n 2 𝐴 = 5 7 6 6 2 5 , where 1 8 0 < 𝐴 < 2 7 0 , and t a n 𝐵 = 4 3 , where 9 0 < 𝐵 < 1 8 0 , find s i n ( 𝐴 𝐵 ) .

Q25:

Find the value of s i n ( 𝐵 2 𝐴 ) given t a n 𝐴 = 4 3 where 𝐴 0 , 𝜋 2 and t a n 𝐵 = 2 4 7 where 𝐵 𝜋 , 3 𝜋 2 .

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