Sum and Difference Identities: Videos & Practice Problems

Trigonometric identities are essential tools for simplifying expressions and finding exact values of trigonometric functions. One particularly useful set of identities is the sum and difference identities, which allow us to handle expressions involving the sum or difference of two angles. For instance, when faced with the expression $\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\; +\; \backslash frac\{\backslash pi\}\{6\}\backslash right)$, we can apply the sum identity for sine, which states:

$\backslash sin(a\; +\; b)\; =\; \backslash sin(a)\; \backslash cos(b)\; +\; \backslash cos(a)\; \backslash sin(b)$

In this case, we can let $a\; =\; \backslash frac\{\backslash pi\}\{2\}$ and $b\; =\; \backslash frac\{\backslash pi\}\{6\}$. Substituting these values into the identity gives:

$\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; \backslash cos\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)\; +\; \backslash cos\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; \backslash sin\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)$

From the unit circle, we know that $\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; 1$ and $\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; 0$. Thus, the expression simplifies to:

$1\; \backslash cdot\; \backslash cos\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)\; +\; 0\; \backslash cdot\; \backslash sin\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)\; =\; \backslash cos\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)$

Since $\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{6\}\backslash right)\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}$, we find that:

$\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\; +\; \backslash frac\{\backslash pi\}\{6\}\backslash right)\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}$

In addition to the sine sum identity, there are also identities for cosine. The cosine of the sum of two angles is given by:

$\backslash cos(a\; +\; b)\; =\; \backslash cos(a)\; \backslash cos(b)\; -\; \backslash sin(a)\; \backslash sin(b)$

And for the difference of two angles:

$\backslash cos(a\; -\; b)\; =\; \backslash cos(a)\; \backslash cos(b)\; +\; \backslash sin(a)\; \backslash sin(b)$

These identities are particularly useful when dealing with angles that are not directly on the unit circle. For example, to find $\backslash cos(15^\backslash circ)$, we can express it as the difference of two known angles, such as $45^\backslash circ\; -\; 30^\backslash circ$. Using the cosine difference identity:

$\backslash cos(45^\backslash circ\; -\; 30^\backslash circ)\; =\; \backslash cos(45^\backslash circ)\; \backslash cos(30^\backslash circ)\; +\; \backslash sin(45^\backslash circ)\; \backslash sin(30^\backslash circ)$

Substituting the known values from the unit circle:

$\backslash cos(45^\backslash circ)\; =\; \backslash frac\{\backslash sqrt\{2\}\}\{2\},\; \backslash quad\; \backslash cos(30^\backslash circ)\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{2\},\; \backslash quad\; \backslash sin(45^\backslash circ)\; =\; \backslash frac\{\backslash sqrt\{2\}\}\{2\},\; \backslash quad\; \backslash sin(30^\backslash circ)\; =\; \backslash frac\{1\}\{2\}$

We can rewrite the expression as:

$\backslash frac\{\backslash sqrt\{2\}\}\{2\}\; \backslash cdot\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}\; +\; \backslash frac\{\backslash sqrt\{2\}\}\{2\}\; \backslash cdot\; \backslash frac\{1\}\{2\}\; =\; \backslash frac\{\backslash sqrt\{6\}\}\{4\}\; +\; \backslash frac\{\backslash sqrt\{2\}\}\{4\}\; =\; \backslash frac\{\backslash sqrt\{6\}\; +\; \backslash sqrt\{2\}\}\{4\}$

This process illustrates how sum and difference identities can simplify the evaluation of trigonometric functions, especially for angles that are not standard on the unit circle. By recognizing opportunities to apply these identities, we can effectively find exact values and simplify complex expressions.

In trigonometry, rewriting angles as the sum or difference of known angles on the unit circle can simplify the process of finding exact values for trigonometric functions. This technique is particularly useful for angles that are not commonly memorized, such as 75 degrees or 7π/12 radians.

For example, to find the sine of 75 degrees, we can express it as the sum of two angles that we know. By examining common angles in the first quadrant, such as 30 degrees and 45 degrees, we find that:

30° + 45° = 75°.

Using the sine sum formula, we can then calculate:

sin(75°) = sin(30° + 45°) = sin(30°)cos(45°) + cos(30°)sin(45°).

Similarly, for the cosine of -15 degrees, we can look for two angles that subtract to give us -15 degrees. By using 30 degrees and 45 degrees, we find:

30° - 45° = -15°.

Thus, we can apply the cosine difference formula:

cos(-15°) = cos(30° - 45°) = cos(30°)cos(45°) + sin(30°)sin(45°).

When dealing with radians, such as 7π/12, we can also express this angle as a sum of known angles. The angles π/6, π/4, and π/3 are useful here. Noticing that:

3π/12 + 4π/12 = 7π/12,

we can simplify this to:

π/4 + π/3.

Using the cosine sum formula, we can find:

cos(7π/12) = cos(π/4 + π/3) = cos(π/4)cos(π/3) - sin(π/4)sin(π/3).

In summary, recognizing angles that are multiples of 15 degrees or π/12 radians allows us to effectively use sum and difference identities to compute trigonometric values. This method not only streamlines calculations but also enhances understanding of angle relationships on the unit circle.

To find the exact value of the expression sin(10°) * cos(20°) + sin(20°) * cos(10°), we can utilize the sine sum formula. This expression resembles the sine addition formula, which states:

sin(a) * cos(b) + sin(b) * cos(a) = sin(a + b)

In this case, let a = 10° and b = 20°. Applying the formula, we can rewrite the expression as:

sin(10° + 20°)

Calculating the sum gives us:

sin(30°)

From the unit circle, we know that:

sin(30°) = 1/2

Thus, the exact value of the original expression is 1/2. This approach highlights the importance of recognizing patterns in trigonometric identities, particularly the sum and difference formulas, to simplify complex expressions effectively.

In trigonometry, sum and difference identities are essential tools for simplifying expressions involving angles. These identities allow us to expand and simplify trigonometric functions that include variables. For instance, when working with the expression sin(β + 30°), we can apply the sum formula for sine, which states:

sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

By substituting A = θ and B = 30°, we expand the expression to:

sin(θ)cos(30°) + cos(θ)sin(30°)

From the unit circle, we know that cos(30°) = \frac{\sqrt{3}}{2} and sin(30°) = \frac{1}{2}. Thus, the expression becomes:

sin(θ) \cdot \frac{\sqrt{3}}{2} + cos(θ) \cdot \frac{1}{2}

Rearranging this, we can express it as:

\frac{\sqrt{3}}{2}sin(θ) + \frac{1}{2}cos(θ)

Factoring out the common denominator of 2 gives us:

\frac{1}{2}(\sqrt{3}sin(θ) + cos(θ))

This is the simplified form of the original expression using the sum identity.

Next, consider the expression cos(\frac{\pi}{4} - θ). Here, we utilize the difference formula for cosine, which is defined as:

cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

Substituting A = \frac{\pi}{4} and B = θ, we expand it to:

cos(\frac{\pi}{4})cos(θ) + sin(\frac{\pi}{4})sin(θ)

Knowing that both cos(\frac{\pi}{4}) and sin(\frac{\pi}{4}) equal \frac{\sqrt{2}}{2}, we can rewrite the expression as:

\frac{\sqrt{2}}{2}cos(θ) + \frac{\sqrt{2}}{2}sin(θ)

Factoring out the common coefficient results in:

\frac{\sqrt{2}}{2}(cos(θ) + sin(θ))

This demonstrates how to effectively use sum and difference identities to simplify trigonometric expressions involving variables, enhancing your understanding and application of these fundamental concepts in trigonometry.

Understanding the sum and difference identities for tangent is essential for simplifying expressions and finding exact values of trigonometric functions. The identities for tangent are slightly more complex than those for sine and cosine, but they serve a similar purpose.

The tangent of the sum of two angles, \( a \) and \( b \), is given by the formula:

$$ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} $$

Conversely, the tangent of the difference of two angles is expressed as:

$$ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} $$

In these formulas, notice that for the sum, we add in the numerator and subtract in the denominator, while for the difference, we subtract in the numerator and add in the denominator.

To illustrate these identities, consider the example of finding \( \tan\left(\pi + \frac{\pi}{4}\right) \). Using the sum identity, we expand it to:

$$ \tan\left(\pi + \frac{\pi}{4}\right) = \frac{\tan \pi + \tan\left(\frac{\pi}{4}\right)}{1 - \tan \pi \tan\left(\frac{\pi}{4}\right)} $$

Since \( \tan \pi = 0 \) and \( \tan\left(\frac{\pi}{4}\right) = 1 \), the expression simplifies to:

$$ \tan\left(\pi + \frac{\pi}{4}\right) = \frac{0 + 1}{1 - 0} = 1 $$

Next, consider the expression \( \tan(\theta - 45^\circ) \). Applying the difference identity, we have:

$$ \tan(\theta - 45^\circ) = \frac{\tan \theta - \tan 45^\circ}{1 + \tan \theta \tan 45^\circ} $$

Knowing that \( \tan 45^\circ = 1 \), we can substitute to get:

$$ \tan(\theta - 45^\circ) = \frac{\tan \theta - 1}{1 + \tan \theta} $$

Lastly, when dealing with \( \tan(90^\circ + \theta) \), we encounter an undefined value since \( \tan 90^\circ \) is undefined. In such cases, it is beneficial to rewrite the tangent in terms of sine and cosine:

$$ \tan(90^\circ + \theta) = \frac{\sin(90^\circ + \theta)}{\cos(90^\circ + \theta)} $$

Using the sine and cosine sum identities, we can further simplify this expression without encountering undefined values.

By practicing these identities and applying them to various angles, including those that are multiples of \( 15^\circ \) or \( \frac{\pi}{12} \) radians, you can effectively find exact values for trigonometric functions that are not directly on the unit circle. Mastery of these identities will enhance your ability to solve complex trigonometric problems.

In trigonometry, verifying identities involves simplifying one or both sides of an equation to demonstrate their equality. This process often utilizes fundamental trigonometric identities, including the sum and difference identities. A common approach is to start with the more complex side of the equation, applying relevant identities to simplify it.

For example, consider the identity:

\[ \cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta) \]

To verify this identity, we begin with the left side, which contains a difference in its argument. We can apply the cosine difference formula:

\[ \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \]

In this case, substituting \( a = \frac{\pi}{2} \) and \( b = \theta \), we have:

\[ \cos\left(\frac{\pi}{2} - \theta\right) = \cos\left(\frac{\pi}{2}\right)\cos(\theta) + \sin\left(\frac{\pi}{2}\right)\sin(\theta) \]

From the unit circle, we know that:

• \( \cos\left(\frac{\pi}{2}\right) = 0 \)
• \( \sin\left(\frac{\pi}{2}\right) = 1 \)

Substituting these values into the equation yields:

\[ \cos\left(\frac{\pi}{2} - \theta\right) = 0 \cdot \cos(\theta) + 1 \cdot \sin(\theta) = \sin(\theta) \]

Thus, we find that:

\[ \sin(\theta) = \sin(\theta) \]

This confirms that both sides of the equation are equal, successfully verifying the identity. This particular identity is recognized as a cofunction identity, illustrating the relationship between sine and cosine functions.

By practicing these verification techniques using various trigonometric identities, students can deepen their understanding of the relationships between different trigonometric functions and enhance their problem-solving skills in trigonometry.

In this problem, we are tasked with verifying the identity that states the cotangent of \(90^\circ - \theta\) is equal to the tangent of \(\theta\). To begin, we focus on the left side of the equation, which is more complex due to the difference in the argument. We can rewrite the cotangent using its reciprocal identity, which states that \(\cot(\theta) = \frac{1}{\tan(\theta)}\). Thus, we express the left side as:

\[\cot(90^\circ - \theta) = \frac{1}{\tan(90^\circ - \theta)}\]

However, instead of applying the tangent difference formula directly, we recognize that the tangent of \(90^\circ\) is undefined. To avoid this issue, we will express cotangent in terms of sine and cosine. The cotangent can be rewritten as:

\[\cot(90^\circ - \theta) = \frac{\cos(90^\circ - \theta)}{\sin(90^\circ - \theta)}\]

Next, we apply the sum and difference formulas for sine and cosine. For the numerator, we use the cosine difference formula:

\[\cos(90^\circ - \theta) = \cos(90^\circ)\cos(\theta) + \sin(90^\circ)\sin(\theta)\]

Knowing that \(\cos(90^\circ) = 0\) and \(\sin(90^\circ) = 1\), this simplifies to:

\[\cos(90^\circ - \theta) = 0 \cdot \cos(\theta) + 1 \cdot \sin(\theta) = \sin(\theta)\]

For the denominator, we apply the sine difference formula:

\[\sin(90^\circ - \theta) = \sin(90^\circ)\cos(\theta) - \cos(90^\circ)\sin(\theta)\]

Again substituting the known values, we find:

\[\sin(90^\circ - \theta) = 1 \cdot \cos(\theta) - 0 \cdot \sin(\theta) = \cos(\theta)\]

Now substituting these results back into our expression for cotangent, we have:

\[\cot(90^\circ - \theta) = \frac{\sin(\theta)}{\cos(\theta)}\]

This simplifies to:

\[\cot(90^\circ - \theta) = \tan(\theta)\]

Thus, we have verified the identity, confirming that the left side equals the right side:

\[\cot(90^\circ - \theta) = \tan(\theta)\]

This identity is recognized as one of the cofunction identities, illustrating the relationship between cotangent and tangent through complementary angles. If you have any questions about this process, feel free to ask!

To verify the trigonometric identity given by the equation:

\[\frac{\cos(a + b)}{\cos(a) \cos(b)} = -\tan(a) \tan(b) + 1\]

we start with the left side, which appears more complex. Using the cosine sum formula, we can expand the numerator:

\[\cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b)\]

Substituting this into our equation gives us:

\[\frac{\cos(a) \cos(b) - \sin(a) \sin(b)}{\cos(a) \cos(b)}\]

Next, we can separate this fraction into two parts:

\[\frac{\cos(a) \cos(b)}{\cos(a) \cos(b)} - \frac{\sin(a) \sin(b)}{\cos(a) \cos(b)}\]

This simplifies to:

\[1 - \frac{\sin(a)}{\cos(a)} \cdot \frac{\sin(b)}{\cos(b)}\]

Recognizing that \(\frac{\sin(a)}{\cos(a)}\) is \(\tan(a)\) and \(\frac{\sin(b)}{\cos(b)}\) is \(\tan(b)\), we rewrite the expression as:

\[1 - \tan(a) \tan(b)\]

Now, we can rearrange this to match the right side of the original equation:

\[1 - \tan(a) \tan(b) = -\tan(a) \tan(b) + 1\]

Both sides of the equation are now equal, confirming that the identity is verified:

\[\frac{\cos(a + b)}{\cos(a) \cos(b)} = -\tan(a) \tan(b) + 1\]

This process illustrates the importance of recognizing and applying trigonometric identities, such as the cosine sum formula and the definitions of tangent, to simplify and verify equations effectively.

To verify the identity involving trigonometric functions, we start with the equation:

\[\frac{\sin(a - b)}{\tan(a) \tan(b)} = \cos(a) \cos(b) \left( \cot(b) - \cot(a) \right)\]

We will simplify the left side, which contains the fraction, as it is generally more complex. Using the sine difference formula, we expand the numerator:

\[\sin(a - b) = \sin(a) \cos(b) - \cos(a) \sin(b)\]

This gives us:

\[\frac{\sin(a) \cos(b) - \cos(a) \sin(b)}{\tan(a) \tan(b)}\]

Next, we express the tangents in terms of sine and cosine:

\[\tan(a) = \frac{\sin(a)}{\cos(a)}, \quad \tan(b) = \frac{\sin(b)}{\cos(b)}\]

Substituting these into the denominator results in:

\[\tan(a) \tan(b) = \frac{\sin(a)}{\cos(a)} \cdot \frac{\sin(b)}{\cos(b)} = \frac{\sin(a) \sin(b)}{\cos(a) \cos(b)}\]

Now, substituting this back into our expression, we have:

\[\frac{\sin(a) \cos(b) - \cos(a) \sin(b)}{\frac{\sin(a) \sin(b)}{\cos(a) \cos(b)}}\]

To simplify, we multiply by the reciprocal of the denominator:

\[= \left( \sin(a) \cos(b) - \cos(a) \sin(b) \right) \cdot \frac{\cos(a) \cos(b)}{\sin(a) \sin(b)}\]

Breaking this into two fractions gives:

\[\frac{\sin(a) \cos(b) \cos(a) \cos(b)}{\sin(a) \sin(b)} - \frac{\cos(a) \sin(b) \cos(a) \cos(b)}{\sin(a) \sin(b)}\]

Now, we can cancel the common terms:

\[= \cos(a) \cos(b) \left( \frac{\cos(b)}{\sin(b)} - \frac{\cos(a)}{\sin(a)} \right)\]

Recognizing that \(\frac{\cos(b)}{\sin(b)}\) and \(\frac{\cos(a)}{\sin(a)}\) are cotangents, we rewrite this as:

\[= \cos(a) \cos(b) \left( \cot(b) - \cot(a) \right)\]

This matches the right side of our original equation, confirming that:

\[\frac{\sin(a - b)}{\tan(a) \tan(b)} = \cos(a) \cos(b) \left( \cot(b) - \cot(a) \right)\]

Thus, we have successfully verified the identity.

When evaluating the sine of the sum of two angles, \( \sin(a + b) \), using given trigonometric values without specific angle measures, it is essential to follow a systematic approach. The sine sum identity states:

\[\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b)\]

In this scenario, we are provided with \( \cos(a) = \frac{4}{5} \) and \( \sin(b) = \frac{5}{13} \), with angle \( a \) located in the fourth quadrant and angle \( b \) in the second quadrant. The first step involves expanding the sine sum identity and identifying any unknown trigonometric values, which in this case are \( \sin(a) \) and \( \cos(b) \).

Next, sketching right triangles in the appropriate quadrants helps visualize the problem. For angle \( a \) in the fourth quadrant, where \( \cos(a) = \frac{4}{5} \), the adjacent side is 4 and the hypotenuse is 5. The opposite side can be determined using the Pythagorean theorem, yielding \( \sin(a) = \frac{-3}{5} \) (noting that sine is negative in the fourth quadrant).

For angle \( b \) in the second quadrant, where \( \sin(b) = \frac{5}{13} \), the opposite side is 5 and the hypotenuse is 13. The adjacent side can also be found using the Pythagorean theorem, resulting in \( \cos(b) = \frac{-12}{13} \) (since cosine is negative in the second quadrant).

Now that we have all necessary values, we can substitute them back into the sine sum identity:

\[\sin(a + b) = \left(\frac{-3}{5}\right) \left(\frac{-12}{13}\right) + \left(\frac{4}{5}\right) \left(\frac{5}{13}\right)\]

Calculating each term gives:

\[\sin(a + b) = \frac{36}{65} + \frac{20}{65} = \frac{56}{65}\]

Thus, the final result for \( \sin(a + b) \) is \( \frac{56}{65} \). This methodical approach ensures clarity and accuracy when dealing with trigonometric identities and values, even in the absence of specific angle measures.

To find the sine of the sum of two angles, \( a + b \), we can utilize the sine sum identity, which states:

\[\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b)\]

In this problem, we know that \( \sin(a) = \frac{3}{5} \) and \( \sin(b) = -\frac{1}{2} \). We need to determine the cosine values for both angles \( a \) and \( b \) to proceed with the calculation.

First, we identify the quadrants for each angle. Angle \( a \) is in the second quadrant (between \( \frac{\pi}{2} \) and \( \pi \)), where sine is positive and cosine is negative. Angle \( b \) is in the fourth quadrant (between \( \frac{3\pi}{2} \) and \( 2\pi \)), where sine is negative and cosine is positive.

Next, we sketch right triangles for both angles. For angle \( a \), with \( \sin(a) = \frac{3}{5} \), we label the opposite side as 3 and the hypotenuse as 5. Using the Pythagorean theorem:

\[x^2 + 3^2 = 5^2 \implies x^2 + 9 = 25 \implies x^2 = 16 \implies x = -4\]

Thus, \( \cos(a) = -\frac{4}{5} \).

For angle \( b \), with \( \sin(b) = -\frac{1}{2} \), we label the opposite side as -1 and the hypotenuse as 2. Again, applying the Pythagorean theorem:

\[x^2 + (-1)^2 = 2^2 \implies x^2 + 1 = 4 \implies x^2 = 3 \implies x = \sqrt{3}\]

Thus, \( \cos(b) = \frac{\sqrt{3}}{2} \).

Now we can substitute these values back into the sine sum identity:

\[\sin(a + b) = \left(\frac{3}{5}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(-\frac{4}{5}\right) \left(-\frac{1}{2}\right)\]

Calculating each term gives:

\[\sin(a + b) = \frac{3\sqrt{3}}{10} + \frac{4}{10} = \frac{3\sqrt{3} + 4}{10}\]

This expression, \( \frac{3\sqrt{3} + 4}{10} \), represents the sine of the sum of angles \( a \) and \( b \). If you have any questions or need further clarification, feel free to ask!