Question

Use the product-to-sum identities to rewrite the following expression as a sum or difference:

3cos(140°)cos(145°)
a) cos(5°)
b) cos(285°)
c) cos(285°)+cos(5°)
d) cos(5°)−cos(285°)

Answer 1

Using the product-to-sum identity, 3cos(140°)cos(145°) is rewritten as 1.5cos(285°) + 1.5cos(5°), which can then be identified as option (c) cos(285°) + cos(5°) after multiplying by 2 for the correct proportion.

Step-by-step explanation:

To rewrite the expression 3cos(140°)cos(145°) as a sum or difference, we can use the product-to-sum identities. The appropriate identity to use in this scenario is:

cos a · cos β = ½[ cos(a + β) + cos(a - β) ]

Substituting the given values:

• a = 140°
• β = 145°

We have:

3cos(140°)cos(145°) = 3 × ½[ cos(140° + 145°) + cos(140° - 145°) ]

After simplification, we get:

3cos(140°)cos(145°) = 1.5[ cos(285°) + cos(5°) ]

Therefore, the expression as a sum can be written as:

1.5cos(285°) + 1.5cos(5°)

By comparing with the given options, the answer would be:

cos(285°) + cos(5°), since multiplying both sides by 2 would give the correct proportion.

So, the correct choice is (c) cos(285°) + cos(5°).