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Trigonometric Ratios

🔄 Quick Recap​

In the last section, we learned that trigonometry helps us find unknown distances and heights by understanding the relationships between the angles and sides of triangles. Now, let's discover exactly what these relationships are!

📚 Understanding the Parts of a Right Triangle​

Before we learn about trigonometric ratios, we need to know the parts of a right triangle. A right triangle has one angle that is 90° (a right angle).

Parts of a Right Triangle

Looking at the diagram above, we have a right triangle ABC with:

  • Points A, B, and C marking the three corners (vertices) of the triangle
  • Angle B is a right angle (90°), shown by the small square symbol in the corner
  • The three sides of the triangle have special names depending on which angle we're focusing on

When we focus on angle A:

  • The hypotenuse is always the longest side of the right triangle (AC in our diagram). It's always opposite to the right angle.
  • The opposite side is the side directly across from angle A (BC in our diagram). It doesn't touch angle A at all.
  • The adjacent side is the side that helps form angle A, but isn't the hypotenuse (AB in our diagram). It's the side next to angle A.

If we were to focus on angle C instead, the names of the sides would change:

  • The hypotenuse would still be AC (always opposite to the right angle)
  • The opposite side would be AB (directly across from angle C)
  • The adjacent side would be BC (next to angle C, but not the hypotenuse)

This is very important to remember: the names "opposite" and "adjacent" depend on which angle we're working with!

📚 The Six Trigonometric Ratios​

Now, let's define the six trigonometric ratios for an acute angle (an angle less than 90°) in a right triangle.

For angle A in the right triangle ABC:

Primary Trigonometric Ratios​

  1. Sine of angle A (sin A):

    sin A = Opposite side / Hypotenuse = BC/AC

    The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.

  2. Cosine of angle A (cos A):

    cos A = Adjacent side / Hypotenuse = AB/AC

    The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

  3. Tangent of angle A (tan A):

    tan A = Opposite side / Adjacent side = BC/AB

    The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

Secondary Trigonometric Ratios (Reciprocals)​

The remaining three ratios are actually the reciprocals (1 divided by) of the first three:

  1. Cosecant of angle A (cosec A):

    cosec A = 1/sin A = Hypotenuse / Opposite side = AC/BC

    The cosecant is the reciprocal of sine.

  2. Secant of angle A (sec A):

    sec A = 1/cos A = Hypotenuse / Adjacent side = AC/AB

    The secant is the reciprocal of cosine.

  3. Cotangent of angle A (cot A):

    cot A = 1/tan A = Adjacent side / Opposite side = AB/BC

    The cotangent is the reciprocal of tangent.

🧠 Memory Tricks​

To remember the primary trigonometric ratios, you can use the acronym SOH-CAH-TOA. Say it out loud: "SOH-CAH-TOA"

  • Sine = Opposite / Hypotenuse
  • Cosine = Adjacent / Hypotenuse
  • Tangent = Opposite / Adjacent

You can create a sentence to remember this acronym, like "Some Old Hippo Caught A Heart Trying Other Animal"

📚 Relationships Between Trigonometric Ratios​

The six ratios are related to each other in several ways:

  1. The secondary ratios are reciprocals of the primary ones:

    cosec A = 1/sin A
    sec A = 1/cos A
    cot A = 1/tan A

  2. Tangent can be expressed as a ratio of sine and cosine:

    tan A = sin A / cos A

    This is true because:

    tan A = Opposite/Adjacent
    sin A / cos A = (Opposite/Hypotenuse) / (Adjacent/Hypotenuse) = Opposite/Adjacent

  3. Similarly, cotangent can be expressed as:

    cot A = cos A / sin A

These relationships are very useful because they allow us to find all trigonometric ratios if we know just one of them!

⚠️ Common Misconceptions​

  1. Misconception: sin A is the product of "sin" and A. Fact: sin A is a mathematical function, not a product. "sin" is not a separate entity that can be multiplied with A. It's simply the notation for the sine function applied to angle A.

  2. Misconception: The values of trigonometric ratios depend on the size of the triangle. Fact: The trigonometric ratios depend only on the angle, not on the size of the triangle. Similar triangles have the same trigonometric ratios, even if one triangle is much larger than the other.

  3. Misconception: Trigonometric ratios only work for certain special angles. Fact: Trigonometric ratios are defined for any angle, though we'll focus on acute angles (0° to 90°) for now.

✅ Solved Examples​

Let's work through some examples to see how we can use these ratios in practice.

Example 1:​

In a right triangle PQR, right-angled at Q, if tan P = 3/4, find the values of sin P and cos P.

Solution: Given: tan P = 3/4

Let's draw this triangle and label it. Since tan P = Opposite/Adjacent = 3/4, we know the ratio of these sides.

If we let the opposite side = 3k and adjacent side = 4k (where k is some positive number), we can find the hypotenuse using the Pythagorean theorem:

Hypotenuse² = Opposite² + Adjacent²
Hypotenuse² = (3k)² + (4k)²
Hypotenuse² = 9k² + 16k²
Hypotenuse² = 25k²
Hypotenuse = 5k

Now we can find sin P and cos P:

sin P = Opposite/Hypotenuse = 3k/5k = 3/5
cos P = Adjacent/Hypotenuse = 4k/5k = 4/5

Note that we can verify our answer using the relationship tan P = sin P / cos P:

sin P / cos P = (3/5) / (4/5) = 3/4 = tan P ✓

Example 2:​

If sin A = 1/2, find the values of cos A and tan A.

Solution: Given: sin A = 1/2

This means that the opposite side / hypotenuse = 1/2. If we call the opposite side "k" and hypotenuse "2k," then we need to find the adjacent side.

Using the Pythagorean theorem:

Adjacent² = Hypotenuse² - Opposite²
Adjacent² = (2k)² - (k)²
Adjacent² = 4k² - k²
Adjacent² = 3k²
Adjacent = √3 k

Now we can find cos A and tan A:

cos A = Adjacent/Hypotenuse = √3k/2k = √3/2
tan A = Opposite/Adjacent = k/(√3k) = 1/(√3) = √3/3

We can check our answer using sin² A + cos² A = 1:

(1/2)² + (√3/2)² = 1/4 + 3/4 = 1 ✓

🤔 Think About It!​

  1. Why do you think the value of sin A is always between 0 and 1 for any acute angle A? (Hint: Think about the lengths of the opposite side and hypotenuse as the angle changes)

  2. What happens to the values of sin A and cos A as angle A gets larger and closer to 90°? (Hint: Imagine what happens to the opposite and adjacent sides)

  3. Can you think of a real-life situation where you might need to use tan A? (For example, calculating the height of a tall building when you know the distance from it)

🔜 What Next?​

In the next section, we will learn the trigonometric ratios for specific angles that come up frequently in mathematics: 0°, 30°, 45°, 60°, and 90°. Knowing these values will help us solve many problems without calculators!