Trigonometry: Sine, Cosine, Tangent
Table Of Contents
🎬 Math Angel Video: Understand Sin, Cos & Tan in Right Triangles
What are Sin, Cos, and Tan?
⏩️
In a right-angled triangle, sine (sin), cosine (cos), and tangent (tan) are trigonometric ratios that compare two sides relative to a given angle θ:
- Sine (sin θ) is the ratio of the opposite side of angle θ to the hypotenuse. $$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$
- Cosine (cos θ) is the ratio of the adjacent side of angle θ to the hypotenuse. $$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$
- Tangent (tan θ) is the ratio of the opposite side of angle θ to the adjacent side of angle θ. $$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$$
❇️ Key Understanding:
The values of sin θ, cos θ, and tan θ depend only on the angle θ, not the size of the triangle. Even if the triangle gets bigger or smaller, the ratios remain the same as long as the angle θ stays the same.
How to Use Sin to Find a Side?
⏩️
We can use sine to find a missing side in a right-angled triangle. If we are given:
- Angle: $30^\circ$
- Opposite side: $4 \text{ cm}$
🛎️ How to Find the Hypotenuse:
To find the hypotenuse, we use the sine formula: $$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$
Step 1: Plug in the value that you know
$$\sin 30^\circ = \frac{4 \text{ cm}}{\text{Hypotenuse}}$$
Step 2: Substitute the value of $\sin 30^\circ$
$$\frac{1}{2} = \frac{4 \text{ cm}}{\text{Hypotenuse}}$$
Step 3: Rearrange to find the hypotenuse
$$\text{Hypotenuse} = 2 \times 4 \text{ cm} \Rightarrow \text{Hypotenuse} = 8 \text{ cm}$$
✅ Final Answer: The hypotenuse is 8 cm.
How to Use Tan to Find a Side?
⏩️
We can use tangent to find a missing side in a right-angled triangle. If we are given:
- Angle: $50^\circ$
- Adjacent side: $5 \text{ cm}$
🛎️ How to Find the Opposite side:
To find the hypotenuse, we use the tangent formula: $$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$$
Step 1: Plug in the value that you know
$$\tan 50^\circ = \frac{\text{Opposite}}{5 \text{ cm}}$$
Step 2: Substitute the value of $\tan 50^\circ$
$$1.19175 \approx \frac{\text{Opposite}}{5 \text{ cm}}$$
Step 3: Rearrange to find the opposite side
$$\text{Opposite} \approx 5 \text{ cm} \times 1.19175 \Rightarrow \text{Opposite} \approx 6 \text{ cm}$$
✅ Final Answer: The opposite side is around 6 cm.
How to Use Cos to Find an Angle?
⏩️
We can use cosine to find a missing angle in a right-angled triangle. If we are given:
- Adjacent side: $4 \text{ cm}$
- Hypotenuse: $8 \text{ cm}$
🛎️ How to Find the Missing Angle:
To find the angle $\theta$, we use the cosine formula:
$$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$
Step 1: Plug in the value that you know
$$\cos \theta = \frac{4 \text{ cm}}{8 \text{ cm}} = \frac{1}{2}$$
Step 2: Solve for $\theta$ by using the inverse cosine function on a calculator: $$\theta = \cos^{-1}\left(\frac{1}{2}\right)$$
- Press the cos⁻¹ button (If you can’t find it, press SHIFT, then COS)
- Enter $\large \frac{1}{2}$ into the calculator.
- Press =, and the calculator will display 60°
✅ Final Answer: The missing angle is: $\theta = 60^\circ$.
Common Values of Sin, Cos, and Tan
⏩️
Some angles appear so often in trigonometry that it’s best to memorize their exact values.
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| $30^\circ$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ |
| $60^\circ$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{\sqrt{2}}$ | $1$ |
| $45^\circ$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
✅ Exam Tips:
- For 30° and 60°
From a right triangle with sides in the ratio 1 : √3 : 2. - For 45°
From an isosceles right triangle with sides in the ratio 1 : 1 : √2.
🍪 Quiz: Practice Trigonometry with Sin, Cos & Tan
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