Trigonometry is a branch of mathematics that deals with the relationship between angles and their corresponding side lengths in triangles. This chapter introduces the fundamental trigonometric ratios and their applications in solving right-angled triangles.
Basic Trigonometric Ratios:
In a right-angled triangle (ΔABC), where angle C is the right angle (90°):
- Opposite (opp): The side opposite the right angle (side BC).
- Adjacent (adj): The side adjacent to the given angle (side AB for angle A).
- Hypotenuse (hyp): The side opposite the right angle and the longest side (side AC).
There are three primary trigonometric ratios defined for each angle (A, B, or C) in a right triangle:
- Sine (sin): sin A = opposite / hypotenuse = BC / AC
- Cosine (cos): cos A = adjacent / hypotenuse = AB / AC
- Tangent (tan): tan A = opposite / adjacent = BC / AB
Notes:
- These ratios depend on the specific angle and remain constant for a particular angle in a right triangle regardless of the triangle’s size.
- The values of these ratios can be calculated using trigonometric tables or calculators.
Examples:
1. Finding Trigonometric Ratios
Consider a right triangle ΔABC, where angle C is 90°, AB = 12 cm, and AC = 15 cm. Find sin A, cos A, and tan A.
Solution:
- Identify opposite and adjacent sides relative to angle A:
- Opposite (opp) = BC (since A is opposite BC)
- Adjacent (adj) = AB
- Calculate the ratios:
- sin A = BC / AC = opp / hyp = (unknown) / 15
- cos A = AB / AC = adj / hyp = 12 / 15 = 4/5
- tan A = BC / AB = opp / adj = (unknown) / 12
Note: We cannot determine sin A and tan A without additional information about BC. We can either find the length of BC using the Pythagorean theorem or be given its value.
2. Using Trigonometric Ratios to Find Missing Side Lengths
In a right triangle ΔDEF, angle D is 90°, DE = 8 cm, and sin F = 3/4. Find DF.
Solution:
- Identify relevant ratio and side based on the given information:
- sin F = opposite (DE) / hypotenuse (DF) = 3/4 (given)
- Rearrange the formula to isolate DF (the missing side):
- DF = DE / sin F
- Substitute known values and solve:
- DF = 8 cm / (3/4) = 8 cm * (4/3) = 32/3 cm (approximately 10.67 cm)
3. Real-World Application: Ramp Inclination
Imagine a wheelchair ramp inclined at an angle θ (theta) with the horizontal ground. The length of the ramp along the ground (horizontal distance) is 10 meters, and the vertical height it reaches is 2 meters. You can use trigonometry to find the angle of inclination (θ).
Solution:
- Identify relevant ratio and sides:
- The vertical height (2 meters) is the opposite side relative to angle θ.
- The horizontal distance (10 meters) is the adjacent side.
- We need to find tan θ.
- Use the tangent formula and solve for θ:
- tan θ = opposite / adjacent = 2 meters / 10 meters = 1/5
- Use a calculator or trigonometric table to find the angle whose tangent is 1/5 (approximately 11.31°).
Therefore, the ramp is inclined at an angle of approximately 11.31° with the horizontal ground.
Pythagorean Identity
This fundamental identity relates the squares of all three sides in a right triangle.
Formula: sin² θ + cos² θ = 1
Explanation:
This identity holds true for any right triangle. By squaring the definitions of sine and cosine, and using some algebraic manipulations, we can arrive at this equation. It essentially states that the sum of the squares of the ratios involving the opposite and adjacent sides is equal to the square of the ratio involving the hypotenuse.
Example:
In the right triangle ΔABC from Example 1 (AB = 12 cm, AC = 15 cm), verify the Pythagorean identity for angle A.
Solution:
We already found cos A = 4/5 and (given) sin A = BC / AC (unknown value).
- Substitute the known values of cos A and square it: (cos A)² = (4/5)² = 16/25
- Since sin² A + cos² A = 1, solve for sin² A: sin² A = 1 – (cos A)² = 1 – (16/25) = 9/25
Note: Although we couldn’t directly calculate sin A earlier, we can now find its square root: sin A = √(sin² A) = √(9/25) = 3/5
This demonstrates that sin² A + cos² A (which is (9/25) + (16/25)) indeed equals 1, verifying the Pythagorean identity.
Applications in Other Triangle Types
While trigonometry is most straightforward with right triangles, some concepts can be extended to other triangle types using techniques like the Law of Sines and the Law of Cosines (introduced in more advanced courses). These laws allow you to solve for missing angles or side lengths in triangles that are not right-angled.
Applications in Periodic Phenomena
Trigonometry plays a crucial role in studying periodic phenomena like sound waves, light waves, and even planetary motion. These phenomena often involve cyclical variations that can be modeled using trigonometric functions (sine, cosine, etc.). By understanding these functions, scientists and engineers can analyze wave properties, predict behavior, and design systems that interact with these waves.
Basic Trigonometric Ratios (Recap)
- Sine (sin): Represents the ratio of the opposite side (BC) to the hypotenuse (AC) in a right triangle ΔABC (sin A = BC / AC). Imagine “S” for “opposite over Slope” (hypotenuse is considered the slope of the right angle).
- Cosine (cos): Represents the ratio of the adjacent side (AB) to the hypotenuse (AC) (cos A = AB / AC). Think of “C” for “adjacent over Cosine.”
- Tangent (tan): Represents the ratio of the opposite side (BC) to the adjacent side (AB) (tan A = BC / AB). Remember “T” for “opposite over Touch” (referencing the tangent line touching the circle used to define trigonometric ratios).
Short Note: These ratios are unitless since they represent the relative lengths of sides.
Unit Circle and Radian Measure
- While we introduced trigonometric ratios using right triangles, a unit circle provides a more general visualization. This circle has a radius of 1 unit, and angles are measured in radians (0 to 2π) rather than degrees (0 to 360°).
- The coordinates of any point on the unit circle correspond to the cosine (x-coordinate) and sine (y-coordinate) values for a specific angle. This allows us to determine trigonometric ratios for any angle, not just those associated with right triangles.
Additional Trigonometric Identities
In addition to the Pythagorean identity (sin² θ + cos² θ = 1), there are other useful trigonometric identities that relate the ratios to each other. These identities can be derived using algebraic manipulations and the unit circle. Some commonly encountered identities include:
- tan² θ + 1 = sec² θ (where sec θ = 1 / cos θ)
- sin (A + B) = sin A cos B + cos A sin B (Sum identity for sine)
- cos (A + B) = cos A cos B – sin A sin B (Sum identity for cosine)
Applications Beyond Right Triangles
- Law of Sines: This law relates the ratios of side lengths to sine values in any triangle. It allows you to solve for missing angles or side lengths when given some information about the triangle’s sides and angles.
- Law of Cosines: Similar to the Law of Sines, this law relates side lengths and cosines in any triangle, enabling you to solve for missing elements when some measurements are known.
Real-World Applications
- Navigation: Trigonometric ratios are used in navigation to determine positions using angles and distances measured between celestial bodies or landmarks.
- Engineering: Trigonometry plays a vital role in structural analysis, calculating forces in beams and trusses, and designing various mechanical components.
- Surveying: In surveying, trigonometry is used to measure distances and angles to map locations and determine elevations.
- Computer Graphics: Creating realistic 3D graphics in games and animations heavily relies on trigonometric calculations for object transformations, lighting effects, and camera positioning.
Short Note: Trigonometry has applications in various fields beyond these examples, demonstrating its versatility as a mathematical tool.
By understanding these additional points and exploring the applications mentioned above, you gain a deeper appreciation for the power and reach of trigonometry in various scientific and practical contexts. Remember, practice is key to mastering these concepts. Look for opportunities to apply trigonometric principles to solve problems and solidify your understanding.
Summary
This chapter provided an introduction to trigonometry, focusing on right triangles. You learned about the basic trigonometric ratios (sine, cosine, tangent), their applications in finding missing side lengths, and the Pythagorean identity. We briefly touched upon applications in other triangle types and periodic phenomena, highlighting the versatility of trigonometry in various fields. Remember that trigonometry extends beyond right triangles, and future chapters will explore more advanced concepts and applications.
