Sum and Difference Identities: Videos & Practice Problems

In trigonometry, sum and difference identities are essential tools for simplifying expressions involving multiple angles. These identities allow us to break down complex trigonometric functions into simpler components, making it easier to evaluate them without a calculator, even for angles not found on the unit circle.

For instance, consider the expression $\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\; +\; \backslash frac\{\backslash pi\}\{6\}\backslash right)$. To simplify this, 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 our case, 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:

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

Similarly, the sine difference identity is given by:

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

For cosine, the sum and difference identities are:

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

and

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

These identities are particularly useful when dealing with angles that are multiples of 15 degrees or $\backslash frac\{\backslash pi\}\{12\}$ radians. For example, to find $\backslash cos(15^\backslash circ)$, we can express it as $\backslash cos(45^\backslash circ\; -\; 30^\backslash circ)$. Using the cosine difference identity, we have:

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

Substituting known values from the unit circle, we find:

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

Thus, the expression becomes:

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

By mastering these sum and difference identities, students can confidently tackle a wide range of trigonometric problems, simplifying complex expressions and finding exact values for angles that are not directly on the unit circle.

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 the first quadrant angles—30 degrees, 45 degrees, and 60 degrees—we can see that:

30° + 45° = 75°.

Thus, we can use the sine sum formula:

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

where A = 30° and B = 45°.

In another case, to find 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°.

We can then apply the cosine difference formula:

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

where A = 30° and B = 45°.

When dealing with radians, such as 7π/12, we can also break it down into known angles. The angles π/6, π/4, and π/3 can be used here. Noticing that:

3π/12 + 4π/12 = 7π/12,

we can simplify this to:

π/4 + π/3.

Using the cosine sum formula again, we can find the cosine of 7π/12:

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

where A = π/4 and B = 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 + b) = sin(a) * cos(b) + cos(a) * sin(b)

In our case, we can identify a as 10 degrees and b as 20 degrees. Therefore, 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 sine addition formula, 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 expanding the expression for the sine of an angle plus a specific degree, we can apply the sine sum formula:

For an angle β plus 30 degrees, the sine can be expressed as:

\[\sin(\beta + 30^\circ) = \sin(\beta) \cos(30^\circ) + \cos(\beta) \sin(30^\circ)\]

Knowing the values from the unit circle, we find that:

• \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)
• \(\sin(30^\circ) = \frac{1}{2}\)

Substituting these values into the expanded expression gives:

\[\sin(\beta + 30^\circ) = \sin(\beta) \cdot \frac{\sqrt{3}}{2} + \cos(\beta) \cdot \frac{1}{2}\]

This can be rearranged to:

\[\sin(\beta + 30^\circ) = \frac{\sqrt{3}}{2} \sin(\beta) + \frac{1}{2} \cos(\beta)\]

Factoring out the common denominator results in:

\[\sin(\beta + 30^\circ) = \frac{1}{2} \left( \sqrt{3} \sin(\beta) + \cos(\beta) \right)\]

Next, consider the cosine of an angle minus another angle, such as \(\frac{\pi}{4} - \theta\). The cosine difference formula is applied as follows:

\[\cos\left(\frac{\pi}{4} - \theta\right) = \cos\left(\frac{\pi}{4}\right) \cos(\theta) + \sin\left(\frac{\pi}{4}\right) \sin(\theta)\]

Again, using known values:

• \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)
• \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)

Substituting these values yields:

\[\cos\left(\frac{\pi}{4} - \theta\right) = \frac{\sqrt{2}}{2} \cos(\theta) + \frac{\sqrt{2}}{2} \sin(\theta)\]

Factoring out the common coefficient results in:

\[\cos\left(\frac{\pi}{4} - \theta\right) = \frac{\sqrt{2}}{2} \left( \cos(\theta) + \sin(\theta) \right)\]

Through these examples, we see how sum and difference identities facilitate the simplification of trigonometric expressions involving variables, making them easier to work with in various mathematical contexts.

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 when adding angles, the numerator involves addition while the denominator involves subtraction. For the difference of angles, the numerator involves subtraction and the denominator involves addition.

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

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

Knowing that \( \tan \pi = 0 \) 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 \( \tan(\theta - 45^\circ) \). Using the difference identity, we have:

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

Since \( \tan 45^\circ = 1 \), this simplifies to:

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

Another important case is when dealing with \( \tan(90^\circ + \theta) \). Here, we rewrite it using sine and cosine to avoid undefined values:

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

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

In summary, mastering the sum and difference identities for tangent allows for effective simplification of trigonometric expressions, especially when dealing with angles that are multiples of \( 15^\circ \) or \( \frac{\pi}{12} \) radians. Practice applying these identities to various problems to enhance your understanding and proficiency.

In trigonometry, verifying identities involves simplifying one or both sides of an equation to demonstrate that they are equal. This process often utilizes fundamental trigonometric identities, including 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 is more complicated due to the difference in the argument. We can apply the cosine difference formula:

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

In this case, let \( a = \frac{\pi}{2} \) and \( b = \theta \). Thus, 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 gives:

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

Now, we compare this result with the right side of the original equation, which is already \( \sin(\theta) \). Since both sides are equal, we have successfully verified the identity:

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

This identity is an example of a cofunction identity, illustrating the relationship between sine and cosine functions. By practicing these verification techniques using sum and difference identities, students can deepen their understanding of trigonometric relationships 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, leading us to express it as:

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

However, using the tangent difference formula directly could lead to an undefined value since \(\tan(90^\circ)\) is undefined. To avoid this, we will express cotangent in terms of sine and cosine. Recall that:

\[\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\]

Thus, we can rewrite the left side 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 have:

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

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 use 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 that the left side equals the right side of the original identity:

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

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 is the more complex expression. 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 \(\frac{\sin(x)}{\cos(x)}\) is the definition of the tangent function, 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 definition 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 us:

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

Now, we can cancel the sine 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 that:

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

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 an opposite side of -3 (since it is negative in the fourth quadrant). Thus, \( \sin(a) = \frac{-3}{5} \).

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 an adjacent side of -12 (negative in the second quadrant). Therefore, \( \cos(b) = \frac{-12}{13} \).

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, we find:

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

Thus, the final result for \( \sin(a + b) \) is \( \frac{56}{65} \). This methodical approach, involving the use of trigonometric identities, right triangle properties, and careful attention to the signs based on quadrant locations, is crucial for solving these types of problems effectively.

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.

Given that angle \( a \) is in the second quadrant (between \( \frac{\pi}{2} \) and \( \pi \)), and angle \( b \) is in the fourth quadrant (between \( \frac{3\pi}{2} \) and \( 2\pi \)), we can sketch right triangles for both angles to find the missing side lengths and subsequently the cosine values.

For angle \( a \), with \( \sin(a) = \frac{3}{5} \), we can label the opposite side as 3 and the hypotenuse as 5. Using the Pythagorean theorem:

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

Since angle \( a \) is in the second quadrant, the adjacent side is negative, giving us \( \cos(a) = -\frac{4}{5} \).

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:

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

Since angle \( b \) is in the fourth quadrant, the adjacent side is positive, leading to \( \cos(b) = \frac{\sqrt{3}}{2} \).

Now we have all the necessary values:

• \( \sin(a) = \frac{3}{5} \)
• \( \cos(a) = -\frac{4}{5} \)
• \( \sin(b) = -\frac{1}{2} \)
• \( \cos(b) = \frac{\sqrt{3}}{2} \)

Substituting these values into the sine sum identity:

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

Calculating each term:

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

This expression, \( \frac{3\sqrt{3} + 4}{10} \), represents the sine of the sum of angles \( a \) and \( b \). This method illustrates the importance of understanding the unit circle and the properties of trigonometric functions in different quadrants.