Gavin Howe
Trigonometry is a branch of mathematics that deals with triangles, specifically the relationships between their sides and angles. Trigonometric laws and identities are derived from right-angled triangles, and they can be used to solve triangles using the values relating to their sides and angles. The sine law, for example, relates the length of sides to the sine of angles of a triangle and can be used to determine the side lengths or angles of a triangle. The Pythagorean theorem, which states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse, is another example of a trigonometric law. Trigonometric laws have many practical applications, including triangulation in geography and astronomy, and they are also used in electrical and electronics engineering.
| Characteristics | Values |
|---|---|
| When to use trigonometric laws | When solving triangles using the values relating to its sides and angles |
| When you don't have a right triangle, the laws of sines and cosines can be used as they work in any triangle | |
| Trigonometry is used when you need to change degrees to radians, or vice versa | |
| Trigonometry is used to find the distance between landmarks in geography | |
| Trigonometry is used to measure the distance to nearby stars in astronomy | |
| Trigonometry is used in satellite navigation systems | |
| Trigonometry is used in electrical and electronics engineering | |
| Trigonometric laws for sides and angles of a triangle | Sine law: a/sinA = b/sinB = c/sinC |
| Cosine law: c2 = a2 + b^2 - 2ab*cosC | |
| Tangent law: derived from sine law, it gives the relationship between the sides and angles of a triangle | |
| Trigonometric identities | Cofunction identities: show the relationship between complementary angles and the trigonometric functions between the sine and cosine, tangent and cotangent, and secant and cosecant pairs |
| Pythagorean identities: derived from the standard Pythagoras theorem for right-angled triangles | |
| Trigonometric even and odd identities: show whether each trigonometric function is an odd or even function | |
| Sum and difference identities: used to find the value of a trigonometric function at a specific angle in terms of two angles with known trigonometric values | |
| Product to sum identity: used to depict the product of the sine and cosine functions as a sum | |
| Sum to product identity: used to express the sum and difference of sines and cosines as products of their sine and cosine functions |
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What You'll Learn
- When you have two side lengths and one non-contained angle?
- When you have two angles and one side length?
- When you need to find the sine, cosine, tangent, secant, cosecant, and cotangent values?
- When you need to double-check the previous trigonometry laws of a triangle?
- When you need to find the missing angle or side of a triangle?
When you have two side lengths and one non-contained angle
Trigonometry is a branch of mathematics that deals with the sides and angles of right-angled triangles. Trigonometric ratios are used to find the lengths and measures of angles in triangles. When working with trigonometry, it may be helpful to think of the triangle as having a point of reference, usually an angle.
Let's consider an example where we have a right-angled triangle ABC, right-angled at B, with the hypotenuse AC = 5cm, base BC = 3cm, and perpendicular AB = 4cm. If we want to find the values of tan θ, sin θ, and cos θ, we can use the trigonometric ratios. We know that tan θ is the ratio of the opposite side to the adjacent side, so tan θ = AB/BC = 4/3. Similarly, we can find sin θ and cos θ using their respective ratios.
In addition to the primary trigonometric ratios, there are three other ratios called the cosecant, secant, and cotangent. These are the multiplicative inverses of sine, cosine, and tangent, respectively. The cosecant is the ratio of the hypotenuse to the opposite side, the secant is the ratio of the hypotenuse to the adjacent side, and the cotangent is the ratio of the adjacent side to the opposite side.
By memorizing the basic trigonometric ratios and their relationships, trigonometry becomes easier and more accessible, especially when dealing with more complex material.
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When you have two angles and one side length
Trigonometry is a branch of mathematics that deals with triangles, specifically the relationships between their lengths and angles. Trigonometric laws and identities are derived from right-angled triangles.
> $ \frac{sinA}{a} = \frac{sinB}{b} = \frac{sinC}{c}$
> or
> $ \frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC}$
Here, a, b, and c are the sides of the triangle, and A, B, and C are the angles. To apply the sine law, you need to be given either two side lengths and one non-contained angle, or two angles and one side length. A non-contained angle in a triangle is one that cannot be between the two given side lengths.
For example, if you know the length of the hypotenuse and one angle (not the right angle), you can use trigonometric functions to find the length of the side opposite the angle. You can multiply the hypotenuse by sin(θ) to get the length of the side opposite the angle. Alternatively, you can multiply the hypotenuse by cos(θ) to get the side adjacent to the angle.
If you have the non-hypotenuse side adjacent to the angle, you can divide it by cos(θ) to get the length of the hypotenuse. You can also multiply this length by tan(θ) to get the length of the side opposite the angle.
The Pythagorean theorem is another important concept in trigonometry, specific to right triangles. It states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.
Other trigonometric laws include the law of cosines, Mollweide's formula, and the law of tangents.
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When you need to find the sine, cosine, tangent, secant, cosecant, and cotangent values
Trigonometry is a branch of mathematics that deals with triangles, specifically the relationships between their angles and side lengths. Trigonometric laws and identities are derived from right-angled triangles, and they can be used to solve problems involving the angles and sides of triangles.
The sine law, or the law of sines, is used to determine the side lengths or angles of a triangle. It is represented as:
$$\frac{sinA}{a} = \frac{sinB}{b} = \frac{sinC}{c}$$ or $$\frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC}$$
To use the sine law to solve for a missing value, we must be given two side lengths and one non-contained angle, or two angles and one side length. The sine law is useful when we know two angles and one side of a triangle and want to find the sides of the triangle. It can also be used to find the area of an oblique triangle.
The cosine law, or the law of cosines, establishes a relationship between the sides and one angle of a triangle. It is represented as:
$$c^2 = a^2 + b^2 - 2ab \cdot cos(C)$$
The tangent law is derived from the sine law and provides a relationship between the sides and angles of a triangle. The Pythagorean theorem, which states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse, is also fundamental in trigonometry.
The reciprocal identities rule defines the reciprocals of the three primary trigonometric ratios: sine, cosine, and tangent. These identities are used to solve various trigonometry problems. The identities for the tangent and cotangent functions can be derived from similar formulas used for the sine and cosine functions. The Pythagorean identities, as the name suggests, are derived from the standard Pythagoras theorem for right-angled triangles.
Cofunction identities show the relationship between complementary angles and the trigonometric functions that exist between the sine and cosine, tangent and cotangent, and secant and cosecant pairs. The angle sum property of a right-angled triangle is used to derive these identities. The sum and difference identities are used to find the value of a trigonometric function at a specific angle in terms of two angles with known trigonometric values. The product-to-sum identity expresses the product of the sine and cosine functions as a sum, and the sum-to-product identity expresses the sum and difference of sines and cosines as products of their sine and cosine functions.
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When you need to double-check the previous trigonometry laws of a triangle
Trigonometry is the study of triangles and the relationships between their angles and side lengths. Trigonometric laws and identities are derived from right-angled triangles, where the height and base are known. Using trigonometric formulas, we can find the sine, cosine, tangent, secant, cosecant, and cotangent values. These formulas are also used to shift angles by π/2, π, 2π, etc.
The sine law is used to determine the side lengths or angles of a triangle when given two side lengths and one non-contained angle, or two angles and one side length. The law of sines formula relates the lengths of the sides of a triangle to the sines of consecutive angles and can be represented as:
$$\frac{sinA}{a} = \frac{sinB}{b} = \frac{sinC}{c}$$ or $$\frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC}$$
The cosine law proves a relationship between the sides and one angle of a triangle and can be represented as:
$$c^2 = a^2 + b^2 - 2ab \cdot cos(C)$$
The tangent law is derived from the sine law and gives the relationship between the sides and angles of a triangle.
Mollweide's formula is another equation that defines the relationship between the sides and angles of a triangle and can be used to double-check previous trigonometry laws. Cofunction identities show the relationship between complementary angles and trigonometric functions, such as sine and cosine, tangent and cotangent, and secant and cosecant. The Pythagorean identities, derived from the Pythagorean theorem for right-angled triangles, are another important set of identities.
In summary, when double-checking previous trigonometry laws of a triangle, one can use Mollweide's formula, cofunction identities, Pythagorean identities, or the sine, cosine, and tangent laws to verify the relationships between the sides and angles of the triangle.
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When you need to find the missing angle or side of a triangle
Trigonometric ratios are used to find the missing sides or angles in a right-angled triangle. The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan), which are derived from the ratios of the sides of a right-angled triangle. For example, the sine function is the ratio of the length of the side opposite an angle to the length of the hypotenuse. The cosine function is the ratio of the length of the side adjacent to an angle to the length of the hypotenuse. The tangent function is the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.
These ratios can be used to determine the ratios of any two sides out of a total of three sides of a right-angled triangle in terms of the respective angles. For instance, if we know the length of one side and the measure of one acute angle in a right triangle, we can use trigonometric ratios to find the length of another side. Similarly, if we know the lengths of two sides of a right triangle, we can use trigonometric ratios to find the measure of the acute angle between those sides.
The six trigonometric ratios are sine, cosine, tangent, cotangent (cot), cosecant (cosec), and secant (sec). These ratios can be calculated using the measure of an acute angle in the right-angled triangle. The values of these trigonometric ratios can be used to find the unknown sides or angles in the triangle. For example, if we know the length of the hypotenuse and one other side of a right triangle, we can use the trigonometric ratios to find the measures of the angles in the triangle, and then use those angles to find the length of the remaining side.
Trigonometry is a branch of mathematics that is particularly useful for finding heights and distances. By applying trigonometric ratios, we can solve problems involving right-angled triangles and gain insights into various mathematical principles.
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