Chapter 7: Q. 16 (page 485)

Find the exact value of the expression:
$\sin 105^{\circ}$

Short Answer

Expert verified

$\frac{1}{4} (\sqrt{6} + \sqrt{2})$

Step by step solution

Step 1. Given information

We have been given an expression $\sin 105^{\circ}$ .

We have to find its exact value.

Step 2. Simplify the expression

We can also write the expression as:

$\sin 105^{\circ} = \sin (135 ° - 30^{°})$

Step 3. Apply the compound angle formula

$\sin (135 ° - 30^{°})$ has the form role="math" $\sin (α - β)$

Therefore, apply

$\sin (α + β) = \sin α \cos β - \cos α \sin β$

localid="1646471429976" $\begin{aligned} \sin 135^{\circ} \cos 30^{\circ} - \cos 135^{\circ} \sin 30^{\circ} = ((\frac{\sqrt{2}}{2}) \cdot (\frac{\sqrt{3}}{2})) - ((- \frac{\sqrt{2}}{2}) \cdot (\frac{1}{2})) \\ = (\frac{\sqrt{6}}{4}) - (- \frac{\sqrt{2}}{4}) \\ = (\frac{\sqrt{6} + \sqrt{2}}{4}) \\ = \frac{1}{4} (\sqrt{6} + \sqrt{2}) \end{aligned}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Find the exact value of the expression:
$\sin 105^{\circ}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Simplify the expression

Step 3. Apply the compound angle formula

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Logic and Functions

Statistics

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Find the exact value of the expression:sin105∘

Short Answer

Step by step solution

Step 1. Given information

Step 2. Simplify the expression

Step 3. Apply the compound angle formula

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Logic and Functions

Statistics

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Find the exact value of the expression:
$\sin 105^{\circ}$