Problem 67 - Algebra and Trigonometry

Question

\(67-72\). Find the value of the product or sum. $$ 2 \sin 52.5^{\circ} \sin 97.5^{\circ} $$

Step-by-Step Solution

Answer

\(\frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\)

Step 1: Use Trigonometric Identity

To find the value of the expression \(2 \sin A \sin B\), we can use the identity: \[ 2 \sin A \sin B = \cos(A-B) - \cos(A+B) \] Here, \(A = 52.5^\circ\) and \(B = 97.5^\circ\).

Step 2: Calculate A - B

Compute \(A - B\) as follows: \[ 52.5^\circ - 97.5^\circ = -45^\circ \] So, \(A - B = -45^\circ\).

Step 3: Calculate A + B

Compute \(A + B\) as follows: \[ 52.5^\circ + 97.5^\circ = 150^\circ \] Thus, \(A + B = 150^\circ\).

Step 4: Substitute in Identity

Now substitute the values found in Steps 2 and 3 into the identity: \[ 2 \sin 52.5^\circ \sin 97.5^\circ = \cos(-45^\circ) - \cos(150^\circ) \] Recall that \(\cos(-\theta) = \cos(\theta)\), thus \(\cos(-45^\circ) = \cos(45^\circ)\).

Step 5: Calculate Cosines

Find the values of the cosines: \[ \cos(45^\circ) = \frac{\sqrt{2}}{2} \] \[ \cos(150^\circ) = -\cos(180^\circ - 150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2} \]

Step 6: Compute Final Result

Substitute these cosine values back into the equation: \[ \cos(45^\circ) - \cos(150^\circ) = \frac{\sqrt{2}}{2} - \left(-\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \] Thus, the final value of the expression is \(\frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics for analyzing angles and their ratios in right-angled triangles. These functions include sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent. Each function gives a specific relationship between the angles and sides of a triangle.

- **Sine (\( \sin \theta \))** measures the ratio of the length of the side opposite the angle to the hypotenuse.- **Cosine (\( \cos \theta \))** measures the ratio of the length of the adjacent side to the hypotenuse.- **Tangent (\( \tan \theta \))** is the ratio of sine to cosine, or the opposite side to the adjacent side.
These functions are not only limited to triangles but are also applicable to circles. They can describe periodic phenomena like sound and light waves. Understanding these functions is crucial for solving problems in physics, engineering, and navigation, as they help relate angles to distances.

Angle Subtraction and Addition

The processes of angle subtraction and addition are vital for solving trigonometric equations using identities. These identities allow us to express complex trigonometric expressions in simpler forms.

For angle addition, the cosine and sine identities are:

• Cosine addition identity: \[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
• Sine addition identity: \[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]

Similarly, angle subtraction identities are:

• Cosine subtraction identity: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
• Sine subtraction identity: \[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]

By using these identities, we can simplify expressions like the one in the exercise: \(2\sin A\sin B = \cos(A-B) - \cos(A+B)\). This allows calculating the subtraction and addition of angles easier, ultimately leading to computational simplicity and accuracy.

Cosine Values

Cosine values represent a crucial trigonometric function in understanding the properties and behaviors of angles and triangles. They exhibit symmetry and periodicity which are useful in trigonometry problems.

Understanding the cosine values for key angles can greatly assist in solving trigonometric expressions. For example:

• \(\cos(0^{\circ}) = 1\)
• \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\)
• \(\cos(45^{\circ}) = \frac{\sqrt{2}}{2}\)
• \(\cos(60^{\circ}) = \frac{1}{2}\)
• \(\cos(90^{\circ}) = 0\)

When solving problems such as the given exercise, it’s necessary to know both positive and negative quadrants of cosine values.Moreover, using identities like \(\cos(-\theta) = \cos(\theta)\) showcases cosine's even function property. Also, for supplementary angles, such as \(\cos(150^{\circ})\) which translates to \(-\cos(30^{\circ})\), understanding change of reference aids in simplifying such problems.