Trigonometric Functions on Right Triangles: Videos & Practice Problems

Trigonometric functions, commonly referred to as trig functions, are essential in understanding the relationships between angles and side lengths in right triangles. The three primary trig functions are sine, cosine, and tangent, each defined as specific ratios of the sides of a right triangle. These functions can be remembered using the mnemonic SOH CAH TOA, which stands for:

• Sine (SOH): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. Mathematically, this is expressed as: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$
• Cosine (CAH): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, represented as: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$
• Tangent (TOA): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side, given by: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$

For example, if we have a right triangle where the length of the opposite side to angle θ is 3 and the hypotenuse is 5, the sine of θ would be calculated as:

Similarly, if the adjacent side is 4, the cosine of θ would be:

And the tangent of θ, which is the ratio of the opposite side to the adjacent side, would be:

Additionally, it is important to note that the tangent function can also be expressed in terms of sine and cosine:

This relationship reinforces the interconnectedness of the trig functions. When solving problems involving different angles, it is crucial to reference the correct angle to determine the appropriate sides. For instance, when calculating the cosine of angle y, one must identify the adjacent side relative to y, which may differ from the adjacent side used for angle x.

In summary, mastering these trigonometric functions and their relationships is vital for success in both current and future mathematical studies. Utilizing the SOH CAH TOA mnemonic can greatly aid in recalling the definitions and applications of these functions in various problems involving right triangles.

In trigonometry, we expand our understanding by introducing three new functions: cosecant, secant, and cotangent. These functions are closely related to the primary trigonometric functions—sine, cosine, and tangent—through what are known as reciprocal identities. Essentially, each of these new functions is the reciprocal of one of the original functions.

The cosecant function is defined as the reciprocal of the sine function. Mathematically, this can be expressed as:

$$ \text{csc}(\theta) = \frac{1}{\sin(\theta)} $$

In terms of a right triangle, if the sine of an angle θ is represented as the ratio of the length of the opposite side to the hypotenuse, then the cosecant is the ratio of the hypotenuse to the opposite side. For example, if the hypotenuse is 5 and the opposite side is 3, then:

$$ \text{csc}(\theta) = \frac{5}{3} $$

Similarly, the secant function is the reciprocal of the cosine function:

$$ \text{sec}(\theta) = \frac{1}{\cos(\theta)} $$

Here, if the cosine of an angle θ is the ratio of the adjacent side to the hypotenuse, then the secant is the ratio of the hypotenuse to the adjacent side. For instance, if the hypotenuse is 13 and the adjacent side is 5, then:

$$ \text{sec}(\theta) = \frac{13}{5} $$

Lastly, the cotangent function is the reciprocal of the tangent function:

$$ \text{cot}(\theta) = \frac{1}{\tan(\theta)} $$

In a right triangle, if the tangent of an angle θ is the ratio of the opposite side to the adjacent side, then the cotangent is the ratio of the adjacent side to the opposite side. For example, if the opposite side is 5 and the adjacent side is 12, then:

$$ \text{cot}(\theta) = \frac{12}{5} $$

These reciprocal identities not only simplify calculations but also enhance our understanding of the relationships between different trigonometric functions. For example, since tangent is defined as the ratio of sine to cosine:

$$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$

It follows that the cotangent can also be expressed as:

$$ \text{cot}(\theta) = \frac{\cos(\theta)}{\sin(\theta)} $$

To effectively utilize these functions, it is crucial to reference the correct angle when applying the identities. By mastering these reciprocal relationships, students can confidently solve various trigonometric problems involving right triangles.

Understanding how to find missing angles in right triangles is essential in trigonometry, and this can be achieved through the use of inverse trigonometric functions. These functions allow us to reverse the process of finding a trigonometric ratio, enabling us to determine the angle when we know the value of a trigonometric function.

To grasp the concept of inverse functions, consider the analogy of exponents. For instance, if we take the number 3 and raise it to the power of 2, we get \(3^2 = 9\). To reverse this operation, we can use the square root, as \(\sqrt{9} = 3\). Similarly, in trigonometry, if we know the sine of an angle, we can find the angle itself by applying the inverse sine function. For example, if the sine of 30 degrees is \( \frac{1}{2} \), then to find the angle from this value, we use the inverse sine: \(\sin^{-1}\left(\frac{1}{2}\right) = 30^\circ\).

Inverse trigonometric functions are also referred to as arc functions. For instance, \(\sin^{-1}(\theta)\) is the same as \(\arcsin(\theta)\). This terminology is important to recognize as you encounter various mathematical problems.

To find a missing angle in a right triangle, follow these steps:

1. **Choose a Trigonometric Function**: Identify the appropriate trigonometric function based on the sides of the triangle relative to the angle in question. The mnemonic SOHCAHTOA can help you remember the relationships: Sine (SOH) relates the opposite side to the hypotenuse.

2. **Write the Equation**: For example, if the opposite side is 6 and the hypotenuse is 13, the equation using sine would be:

\[\sin(\theta) = \frac{6}{13}\]

3. **Apply the Inverse Function**: To isolate the angle, take the inverse sine of both sides:

\[\theta = \sin^{-1}\left(\frac{6}{13}\right)\]

This equation now represents the angle \(\theta\) in terms of the known sides of the triangle. In future applications, you can use a calculator to find the approximate value of this angle.

This introduction to inverse trigonometric functions provides a foundational understanding of how to find missing angles in right triangles, a crucial skill in trigonometry.

Understanding how to evaluate trigonometric functions using a calculator is essential for solving various mathematical problems. Trigonometric functions include sine, cosine, and tangent, while their inverses are represented as arcsine, arccosine, and arctangent. To effectively use a calculator for these functions, it is crucial to ensure that the calculator is set to the correct mode: either degrees or radians. This choice depends on the specific problem being solved.

For example, when calculating the sine of 37 degrees, you must switch your calculator to degree mode. You would input this as \sin(37^\circ), and upon pressing enter, you would obtain approximately 0.6 when rounded to the nearest tenth. Conversely, for the tangent of \frac{2\pi}{15}, you need to be in radian mode. The input would be \tan\left(\frac{2\pi}{15}\right), yielding an approximate value of 0.4 when rounded.

In cases where a specific function like secant is required, remember that secant is the reciprocal of cosine. Therefore, to find the secant of 50 degrees, you would calculate it as \sec(50^\circ) = \frac{1}{\cos(50^\circ)}. After entering this into the calculator, you would round the result to get approximately 1.6.

For inverse functions, such as the arctangent of \frac{3}{4}, you would again ensure the calculator is in degree mode. By pressing the second function key followed by the tangent button, you can input \tan^{-1}\left(\frac{3}{4}\right). This would yield an approximate value of 36.9 degrees when rounded.

In summary, mastering the use of a calculator for trigonometric and inverse trigonometric functions involves understanding the relationship between the functions, knowing when to switch modes, and applying the correct input format. This skill is vital for solving a wide range of mathematical problems effectively.

In this example, we are tasked with finding the missing angle, denoted as θ, in a right triangle using inverse trigonometric functions, specifically the sine function. To begin, it is essential to ensure that your calculator is set to degree mode, as the final answer needs to be expressed in degrees.

We start by identifying the appropriate sides of the triangle to use with the sine function. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. In this case, we have:

$$ \sin(θ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{6}{13} $$

To solve for θ, we apply the inverse sine function:

$$ θ = \sin^{-1}\left(\frac{6}{13}\right) $$

Next, we input this into our calculator. After pressing the second function key followed by the sine key, we enter the ratio 6/13 and close the parenthesis. Upon pressing enter, the calculator will provide the approximate value of θ.

After rounding to two decimal places, we find that:

$$ θ \approx 27.49^\circ $$

This value represents the measure of the missing angle in the triangle, completing our solution. Understanding how to use inverse trigonometric functions is crucial for solving problems involving angles in right triangles.