Sine, cosine, and tangent are mathematical functions commonly used in trigonometry. They are based on the relationships between the sides and angles of a right triangle.
Sine (sin) is the ratio of the length of the side opposite an angle to the length of the hypotenuse. Cosine (cos) is the ratio of the length of the adjacent side to the length of the hypotenuse. Tangent (tan) is the ratio of the length of the opposite side to the length of the adjacent side.
These functions are used to calculate unknown angles or sides of a right triangle when some information is known. They are also used in various fields such as physics, engineering, and navigation to solve problems involving angles and distances.
Many of our Wonder Friends already have knowledge about the Pythagorean Theorem. They may have read about Pascal’s Triangle. They might even have knowledge about the other types of triangles. Today, we are becoming curious about another topic related to triangles. What are we discussing? Sine, cosine, and tangent, obviously!
What are sine, cosine, and tangent? They are three of the main functions in trigonometry. You may have already heard about them in your math class. Trigonometry is associated with geometry and other branches of math. It can assist us in better comprehending the connections between the sides and angles of rectangles.
Sine, cosine, and tangent are significant in the study of right triangles. Have you ever seen this type of triangle? If you have, you know that one of its three angles is always 90° (a right angle). The other two angles can have any measurements, as long as the sum of all three angles is 180°.
How do mathematicians determine sine, cosine, and tangent? They begin with one of the non-right angles in a right triangle. Usually, they label this angle as theta (Θ). Then, they label the three sides of the triangle.
The side of the triangle directly across from the right angle is called the hypotenuse. It is the longest side of the triangle. The side across from theta is referred to as “opposite.” It is easy to remember because it is on the opposite side from the theta angle! Lastly, the side that touches theta but is not the hypotenuse is called “adjacent.”
Correctly labeling the sides is crucial in determining sine, cosine, and tangent. This is because each of them has a formula that divides the length of one side by the length of another. Here are the formulas for determining each of these functions:
Sine Θ = Opposite ÷ Hypotenuse
Cosine Θ = Adjacent ÷ Hypotenuse
Tangent Θ = Opposite ÷ Adjacent
Initially, these formulas may appear difficult to remember. However, memorizing a helpful mnemonic can be beneficial. When determining sine, cosine, and tangent, just remember SOHCAHTOA (sō-kŭ-tō-ŭ). In this case, S, C, and T represent sine, cosine, and tangent. O, A, and H represent Opposite, Adjacent, and Hypotenuse.
By remembering SOHCAHTOA, you will be able to correctly write the formulas for sine, cosine, and tangent. Then, just substitute the correct numbers, and you are ready! Determining these functions is a matter of remembering the formulas and using the appropriate side measurements. Many individuals find it helpful to label the triangle before starting.
Can you determine the sine, cosine, and tangent of an angle? We believe you can! However, new math topics can be challenging at first. With practice and assistance from a teacher or friend, we are confident that you will soon be learning a lot about trigonometry!
Try It Out
Ready to continue learning? Take a look at the activities below with the help of a friend or family member!
- Find out more about sine, cosine, and tangent, and examine some useful diagrams on the Math Is Fun website. Did this assist you in gaining a deeper comprehension of these functions? Summarize what you have learned today for a friend or family member.
- Now that you have a memory aid to assist you in remembering the formulas for sine, cosine, and tangent, create a poster that would aid others in remembering. Utilize the mnemonic SOHCAHTOA, and incorporate any images or diagrams that you believe would assist your friends or family members in learning how to find these functions. Make sure to include the correct formulas as well!
- Feeling artistic? Learn more about geometric shapes and get creative by creating your own! You can attempt to create some shapes on your own or follow along with this video to learn how to create 3D shapes using just paper and a pair of scissors. Make sure to have an adult assist you, and enjoy the process of creating your shapes.
Sources of Wonder
1. What are sine, cosine, and tangent?
Sine, cosine, and tangent are mathematical functions commonly used in trigonometry. They represent the ratios between the sides of a right triangle. Sine (sin) is the ratio of the length of the side opposite the given angle to the hypotenuse. Cosine (cos) is the ratio of the length of the adjacent side to the hypotenuse. Tangent (tan) is the ratio of the length of the opposite side to the adjacent side.
2. How do sine, cosine, and tangent relate to angles?
Sine, cosine, and tangent are used to calculate the values of angles in a right triangle. By using these functions, we can determine the relationship between the lengths of the sides and the measures of the angles. These functions allow us to find missing side lengths or angle measures in a triangle.
3. How are sine, cosine, and tangent calculated?
Sine, cosine, and tangent can be calculated using a calculator or by using trigonometric tables. However, these functions can also be calculated by using the ratios of the sides of a right triangle. For example, to calculate the sine of an angle, divide the length of the side opposite the angle by the hypotenuse.
4. What are the ranges of sine, cosine, and tangent?
The ranges of sine, cosine, and tangent are between -1 and 1. This means that the values of these functions will always fall within this range. The maximum value of sine and cosine is 1, while the minimum value is -1. The range of tangent, however, extends to positive and negative infinity because it can approach these values as the angle approaches 90 degrees.
5. What are the applications of sine, cosine, and tangent?
Sine, cosine, and tangent have various applications in real-life situations. They are used in fields such as engineering, physics, architecture, and navigation. These functions are particularly useful in calculating distances, angles, and heights. For example, they can be used to determine the height of a building or the distance between two objects based on measurements and angles.
6. Can sine, cosine, and tangent be used for angles other than right angles?
Yes, sine, cosine, and tangent can be used for angles other than right angles. While they are commonly associated with right triangles, these functions can also be used for any angle in a triangle. In such cases, they are calculated using the ratios of the sides of the triangle. However, it’s important to note that the values of these functions may differ for non-right angles compared to right angles.