Question

COMPLETE THE SENTENCE In a right triangle, the two trigonometric functions of ? that are defi ned using the lengths of the hypotenuse and the side adjacent to ? are __________ and __________.

Short answer

The two trigonometric functions of ? that are defined using the lengths of the hypotenuse and the side adjacent to ? are 'Cosine' and 'Secant'.

Step-by-step solution

Identifying the functions

For any angle ? in a right-angled triangle, the sides of the triangle are named as follows: \n- The side opposite to the angle ? is called the 'Opposite'\\n- The side adjacent to the angle ? is called the 'Adjacent' \\n- The side that is always the longest is the 'Hypotenuse'. The trigonometric functions that involve hypotenuse and adjacent side are 'Cosine' and 'Secant'.

Explaining the functions

The cosine of the angle ?, often abbreviated as cos(?), is defined as the ratio between the adjacent side and the hypotenuse. In other words, cos(?) = Adjacent / Hypotenuse. The secant of ?, abbreviated as sec(?), is the reciprocal of the cosine. It can be defined as Hypotenuse / Adjacent.

Key concepts

Cosine

The trigonometric function cosine, often symbolized as 'cos', is one of the fundamental concepts students encounter when learning about right triangles and trigonometry. To understand cosine, imagine a right-angled triangle where one of the angles, let's call it \( \theta \), is not the 90-degree angle.

The side that lays next to angle \( \theta \) and isn't the longest side of the triangle is known as the adjacent side. Directly opposite the right angle is the longest side of the triangle, called the hypotenuse. In this context, the cosine of the angle \( \theta \) is simply calculated by dividing the length of the adjacent side by the length of the hypotenuse, mathematically expressed as \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \).

By understanding cosine, students can solve a variety of problems, such as finding missing side lengths in right-angled triangles and determining angles when certain side lengths are known.

Secant

Another important trigonometric function related to right triangles is secant, abbreviated as 'sec'. Secant tends to appear less frequently in basic trigonometry problems but is equally important as cosine.

The secant of an angle \( \theta \) in a right-angled triangle is defined as the reciprocal of the cosine of \( \theta \). This means that instead of dividing the adjacent side by the hypotenuse, for secant, you divide the hypotenuse by the adjacent side. The relationship is \( \sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent}} \), which can also be written as \( \sec(\theta) = \frac{1}{\cos(\theta)} \).

This function is particularly useful in scenarios where the hypotenuse is known and one needs to find the length of the adjacent side. It's also used extensively in higher-level mathematics, including calculus.

Hypotenuse

The hypotenuse is the longest side of a right triangle, and it is always found opposite the right angle. Understanding the hypotenuse is key to tackling trigonometry, as it plays a crucial role in defining trigonometric functions like sine, cosine, and secant.

In the famous Pythagorean theorem, the hypotenuse's length is found using the lengths of the other two sides (\text{opposite} and \text{adjacent}), with the formula \( c = \sqrt{a^2 + b^2} \), where 'c' is the hypotenuse, while 'a' and 'b' are the legs of the triangle.

In trigonometric terms, the hypotenuse serves as a reference side for measuring the ratios that define the sine, cosine, and secant functions. Without the hypotenuse, these calculations would not be possible.

Adjacent side

In the context of a right triangle, the adjacent side is the side that is next to a given angle \( \theta \) and is not the hypotenuse. This side is often the 'starting point' for defining certain trigonometric ratios.

For the cosine function, we use the adjacent side as the numerator when it's divided by the hypotenuse. Similarly, for secant, the adjacent side is placed as the denominator when the hypotenuse is divided by it.

The length of the adjacent side can be determined through various methods if other measurements of the triangle are known, such as using the Pythagorean theorem or through trigonometric functions when the angles are known. It's a key side in solving many problems related to right-angled triangles and helps in understanding the relationship between the sides and angles of the triangle.