Trigonometric Functions on Right Triangles: Videos & Practice Problems

Trigonometric functions, including sine, cosine, and tangent, relate angles to side lengths in right triangles. These functions can be remembered using the mnemonic SOHCAHTOA: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. Additionally, reciprocal identities introduce cosecant, secant, and cotangent, which are the inverses of sine, cosine, and tangent, respectively. Inverse trigonometric functions allow for finding missing angles, utilizing operations like inverse sine to reverse the trigonometric function. Understanding these concepts is essential for solving various mathematical problems involving triangles.

Introduction to Trigonometric Functions

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 opposite side to angle θ measures 3 units and the hypotenuse measures 5 units, the sine of θ would be calculated as:\[\sin(\theta) = \frac{3}{5}\]For the cosine, if the adjacent side measures 4 units, it would be:\[\cos(\theta) = \frac{4}{5}\]And for the tangent, using the opposite and adjacent sides, we find:\[\tan(\theta) = \frac{3}{4}\]

Additionally, it is important to note that the tangent function can also be expressed in terms of sine and cosine:\[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]This relationship reinforces the interconnectedness of these functions.

When solving problems involving trigonometric functions, it is crucial to reference the correct angle. For instance, if calculating the cosine of angle y in a triangle, one must identify the adjacent side relative to angle y, not angle x. This attention to detail ensures accurate calculations.

In summary, mastering the definitions and relationships of sine, cosine, and tangent through the SOH CAH TOA mnemonic is vital for success in trigonometry and future mathematical studies.

Given the right triangle below, evaluate $\cos\theta$.

• A. $\cos\theta=\frac{8}{17}$
• B. $\cos\theta=\frac{8}{15}$
• C. $\cos\theta=\frac{15}{17}$
• D. $\cos\theta=\frac{15}{8}$

Answer: (C)

Given the right triangle below, evaluate $\tan\theta$.

• A. $\tan\theta=\frac35$
• B. $\tan\theta=\frac45$
• C. $\tan\theta=\frac43$
• D. $\tan\theta=\frac34$

Answer: (D)

Fundamental Trigonometric Identities

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 opposite side measures 3 and the hypotenuse measures 5, 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 adjacent side is 4 and the hypotenuse is 5, then:

$$\text{sec}(\theta) = \frac{5}{4}$$

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

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

In this case, 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 3 and the adjacent side is 4, then:

$$\text{cot}(\theta) = \frac{4}{3}$$

These reciprocal identities not only simplify calculations but also help in understanding the relationships between the different trigonometric functions. For instance, 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 apply these concepts, consider a right triangle where you need to find the secant of an angle. If the adjacent side is 5 and the hypotenuse is 13, the cosine would be:

$$\cos(x) = \frac{5}{13}$$

Thus, the secant would be:

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

For the cosecant, if the opposite side is 12 and the hypotenuse is 13, then:

$$\sin(x) = \frac{12}{13}$$

So, the cosecant would be:

$$\text{csc}(x) = \frac{13}{12}$$

Finally, if you need to find the cotangent of another angle in the same triangle, and the opposite side is 5 while the adjacent side is 12, then:

$$\tan(y) = \frac{5}{12}$$

Thus, the cotangent would be:

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

Understanding these reciprocal identities is crucial for solving various trigonometric problems and reinforces the interconnectedness of the trigonometric functions.

If $\tan\theta=\frac{12}{5}$, find the values of the five other trigonometric functions. Rationalize the denominators if necessary.

• A. $\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\cot\theta=\frac{5}{12},\sec\theta=\frac{13}{5},\csc\theta=\frac{13}{12}$
• B. $\sin\theta=\frac{5}{13},\cos\theta=\frac{12}{13},\cot\theta=\frac{5}{12},\sec\theta=\frac{13}{12},\csc\theta=\frac{13}{5}$
• C. $\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\cot\theta=-\frac{5}{12},\sec\theta=-\frac{13}{5},\csc\theta=-\frac{13}{12}$
• D. $\sin\theta=\frac{5}{13},\cos\theta=\frac{12}{13},\cot\theta=-\frac{5}{12},\sec\theta=-\frac{13}{12},\csc\theta=-\frac{13}{5}$

Answer: (A)

If $\sin\theta=\frac{\sqrt{17}}{17}$, find the values of the five other trigonometric functions. Rationalize the denominators if necessary.

• A. $\cos\theta=\frac{\sqrt{17}}{4},\tan\theta=\frac14,\cot\theta=4,\sec\theta=\sqrt{17},\csc\theta=\frac{4\sqrt{17}}{17}$
• B. $\cos\theta=\frac{\sqrt{17}}{4},\tan\theta=-\frac14,\cot\theta=-4,\sec\theta=\sqrt{17},\csc\theta=\frac{4\sqrt{17}}{17}$
• C. $\cos\theta=\frac{4\sqrt{17}}{17},\tan\theta=-\frac14,\cot\theta=-4,\sec\theta=\frac{\sqrt{17}}{4},\csc\theta=\sqrt{17}$
• D. $\cos\theta=\frac{4\sqrt{17}}{17},\tan\theta=\frac14,\cot\theta=4,\sec\theta=\frac{\sqrt{17}}{4},\csc\theta=\sqrt{17}$

Answer: (D)

Introduction to Inverse Trig Functions

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 the ratio is known.

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 ratio, 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 you have an angle \( \theta \) with an opposite side of length 6 and a hypotenuse of length 13, you can express this relationship as:

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

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

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

This equation gives you 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.

In summary, inverse trigonometric functions are powerful tools for determining missing angles in right triangles, and understanding their application is crucial for solving various trigonometric problems.

Given the right triangle below, use the sine function to write a trigonometric expression for the missing angle $\theta$.

• A. $\theta=\sin^{-1}\left(\frac{5}{13}\right)$
• B. $\theta=\sin^{-1}\left(\frac{12}{13}\right)$
• C. $\theta=\sin^{-1}\left(\frac{5}{12}\right)$
• D. $\theta=\sin^{-1}\left(\frac{13}{12}\right)$

Answer: (A)

Given the right triangle below, use the cosine function to write a trigonometric expression for the missing angle $\phi$.

• A. $\phi=\cos^{-1}\left(\frac{12}{13}\right)$
• B. $\phi=\cos^{-1}\left(\frac{13}{5}\right)$
• C. $\phi=\cos^{-1}\left(\frac{13}{12}\right)$
• D. $\phi=\cos^{-1}\left(\frac{5}{13}\right)$

Answer: (D)

How to Use a Calculator for Trig Functions

Understanding how to evaluate trigonometric and inverse trigonometric functions using a calculator is essential for solving various mathematical problems. To begin, it's crucial to familiarize yourself with the primary trigonometric functions: sine, cosine, and tangent. These functions can be accessed through specific buttons on your calculator, which typically include options for both degrees and radians.

Before performing any calculations, ensure your calculator is set to the correct mode. This can be done by pressing the mode button, which will allow you to choose between degrees and radians. The choice of mode depends on the problem at hand. For instance, if you need to find the sine of 37 degrees, switch to degree mode. You would then input the function as follows: sin(37) and press enter. The result, rounded to the nearest tenth, would be approximately 0.6.

In contrast, when calculating the tangent of an angle expressed in radians, such as \frac{2\pi}{15}, ensure your calculator is in radian mode. Input the function as tan\left(\frac{2\pi}{15}\right) and press enter to obtain a result of about 0.4, rounded to the nearest tenth.

For functions like secant, which do not have a dedicated button, recall that secant is the reciprocal of cosine. Therefore, to find the secant of 50 degrees, you would calculate it as \sec(50) = \frac{1}{\cos(50)}. After entering this into your calculator, you would find the result to be approximately 1.5, which rounds to 1.6.

When dealing with inverse trigonometric functions, such as arctangent, remember that this is equivalent to the inverse of tangent. To find the arctangent of \frac{3}{4}, press the second button on your calculator followed by the tangent button. Input the fraction as \tan^{-1}\left(\frac{3}{4}\right) and press enter. The result will be approximately 36.9 degrees when rounded to the nearest tenth.

By mastering these steps and understanding the relationship between trigonometric functions and their inverses, you can effectively utilize your calculator to solve a wide range of problems involving trigonometry.

What is a positive value of A in the interval $\left[0\degree,90\degree\right)$ that will make the following statement true? Express the answer in four decimal places.
$\sin A=0.9235$

• A. $22.5568°$
• B. $67.4432°$
• C. $22.4432°$
• D. $33.5438°$

Answer: (B)

What is the positive value of P in the interval $\left[0\degree,90\degree\right)$ that will make the following statement true? Express the answer in four decimal places.
$\cot P=5.2371$

• A. $55.8102°$
• B. $34.1898°$
• C. $10.8102°$
• D. $79.1898°$

Answer: (C)

What is the positive value of $D$ in the interval $\left[0,\frac{\pi}{2}\right)$ that will make the following statement true? Express the answer in four decimal places.
$\sec D=3.2842$

• A. $0.3094$ rad
• B. $1.2614$ rad
• C. $0.4760$ rad
• D. $1.0934$ rad

Answer: (B)

Example 1

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 the calculator is set to degree mode, as the final answer needs to be expressed in degrees.

We start by identifying the sides of the triangle relevant to 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 the ratio of the opposite side (6) to the hypotenuse (13), which can be expressed as:

$$ \sin(θ) = \frac{6}{13} $$

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

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

Next, we input this into the calculator. By pressing the second function key followed by the sine key, we access the inverse sine function. After entering the ratio \( \frac{6}{13} \) and closing the parenthesis, we press enter to compute the value.

The calculator will yield an approximate value for θ, which we round to two decimal places. The result is:

$$ θ \approx 27.49^\circ $$

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.

Here’s what students ask on this topic:

What are the primary trigonometric functions and how are they defined in a right triangle?

The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). They relate the angles of a right triangle to the lengths of its sides. These functions can be remembered using the mnemonic SOHCAHTOA:

• Sine (sin) is defined as the ratio of the length of the opposite side to the hypotenuse: $\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$.
• Cosine (cos) is the ratio of the length of the adjacent side to the hypotenuse: $\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}$.
• Tangent (tan) is the ratio of the length of the opposite side to the adjacent side: $\frac{\mathrm{opposite}}{\mathrm{adjacent}}$.

How do you use the mnemonic SOHCAHTOA to remember trigonometric functions?

SOHCAHTOA is a mnemonic that helps remember the definitions of the primary trigonometric functions:

• SOH: Sine (sin) is Opposite over Hypotenuse: $\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$.
• CAH: Cosine (cos) is Adjacent over Hypotenuse: $\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}$.
• TOA: Tangent (tan) is Opposite over Adjacent: $\frac{\mathrm{opposite}}{\mathrm{adjacent}}$.

This mnemonic helps you quickly recall which sides of the triangle to use for each function.

What are the reciprocal trigonometric functions and how are they related to the primary trigonometric functions?

The reciprocal trigonometric functions are cosecant (csc), secant (sec), and cotangent (cot). They are the reciprocals of the primary trigonometric functions:

• Cosecant (csc) is the reciprocal of sine: $\frac{\mathrm{hypotenuse}}{\mathrm{opposite}}$.
• Secant (sec) is the reciprocal of cosine: $\frac{\mathrm{hypotenuse}}{\mathrm{adjacent}}$.
• Cotangent (cot) is the reciprocal of tangent: $\frac{\mathrm{adjacent}}{\mathrm{opposite}}$.

These functions are useful in various trigonometric calculations and identities.

How do you find a missing angle in a right triangle using inverse trigonometric functions?

To find a missing angle in a right triangle, you can use inverse trigonometric functions. If you know the value of a trigonometric function, you can find the angle by taking the inverse of that function:

• Inverse sine (sin^-1 or arcsin): $\theta =\; sin-1(\frac{\mathrm{opposite}}{\mathrm{hypotenuse}})$.
• Inverse cosine (cos^-1 or arccos): $\theta =\; cos-1(\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}})$.
• Inverse tangent (tan^-1 or arctan): $\theta =\; tan-1(\frac{\mathrm{opposite}}{\mathrm{adjacent}})$.

These functions will give you the angle in degrees or radians, depending on your calculator settings.

How do you use a calculator to evaluate trigonometric functions?

To evaluate trigonometric functions using a calculator, follow these steps:

1. Ensure your calculator is in the correct mode (degrees or radians) based on the problem.
2. For sine, cosine, and tangent, use the respective buttons (sin, cos, tan) and input the angle.
3. For reciprocal functions (csc, sec, cot), use the reciprocal identities: csc = 1/sin, sec = 1/cos, cot = 1/tan.
4. For inverse functions (sin^-1, cos^-1, tan^-1), press the second function button (often labeled as 2nd or shift) followed by the trigonometric function button.

For example, to find sin(37°), ensure the calculator is in degree mode, press sin, enter 37, and hit equals.