Special Right Triangles: Videos & Practice Problems

Understanding special right triangles, particularly the 45-45-90 and 30-60-90 triangles, is essential for solving geometric problems efficiently. In a 45-45-90 triangle, the legs are equal, and the hypotenuse can be calculated as $a^2=2\times \mathrm{leg}^2$. For a 30-60-90 triangle, the relationships are defined as the hypotenuse being twice the shortest leg and the long leg being the shortest leg multiplied by $\sqrt{3}$. Recognizing these patterns simplifies the calculation of trigonometric functions, enhancing problem-solving skills in mathematics.

45-45-90 Triangles

In the study of right triangles, the 45-45-90 triangle is a special case that frequently appears in various mathematical contexts. This triangle is characterized by having two angles of 45 degrees and one right angle, which results in the two legs being of equal length. This property allows for efficient calculations when determining the lengths of the sides.

To find the hypotenuse of a 45-45-90 triangle, one can use the shortcut that involves multiplying the length of one leg by the square root of 2. For instance, if each leg measures 5 units, the hypotenuse can be calculated as:

$$ c = a \sqrt{2} $$

where \( a \) is the length of a leg. Thus, the hypotenuse would be \( 5 \sqrt{2} \).

Alternatively, the Pythagorean theorem can be applied, which states that the sum of the squares of the legs equals the square of the hypotenuse:

$$ a^2 + b^2 = c^2 $$

In this case, if both legs are 5, the equation becomes:

$$ 5^2 + 5^2 = c^2 $$

Calculating this gives:

$$ 25 + 25 = c^2 $$

$$ 50 = c^2 $$

Taking the square root of both sides results in:

$$ c = \sqrt{50} = 5\sqrt{2} $$

This confirms that both methods yield the same result, but the shortcut is significantly quicker.

When solving for unknown sides in a 45-45-90 triangle, if one leg is known, simply multiply it by the square root of 2 to find the hypotenuse. Conversely, if the hypotenuse is known, divide it by the square root of 2 to find the length of each leg. For example, if the hypotenuse is 13, the calculation would be:

$$ \text{Leg} = \frac{13}{\sqrt{2}} $$

To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{2} \):

$$ \text{Leg} = \frac{13\sqrt{2}}{2} $$

Thus, both legs of the triangle would measure \( \frac{13\sqrt{2}}{2} \).

Understanding these properties and shortcuts for 45-45-90 triangles not only simplifies calculations but also enhances problem-solving efficiency in geometry.

Given the triangle below, determine the missing side(s) without using the Pythagorean theorem (make sure your answer is fully simplified).

• A. $x=81$
• B. $x=9\sqrt2$
• C. $x=2\sqrt9$
• D. $x=\sqrt{162}$

Answer: (B)

Without using a calculator, determine all values of P in the interval $\left[0\degree,90\degree\right)$ with the following trigonometric function value.
$\csc P=\sqrt2$

• A. $P=30\degree$ only
• B. $P=45°$ only
• C. $P=60°$ only
• D. $P=30°,60°$

Answer: (B)

Common Trig Functions For 45-45-90 Triangles

In the study of right triangles, the 45-45-90 triangle is particularly significant due to its unique properties and the relationships between its sides and angles. In a 45-45-90 triangle, the two legs are of equal length, and the hypotenuse can be expressed as the length of a leg multiplied by the square root of 2. This relationship allows for quick calculations of side lengths and trigonometric functions.

When exploring trigonometric functions in the context of a 45-45-90 triangle, we start with the sine function, defined as the ratio of the length of the opposite side to the hypotenuse. For a triangle with legs of length 5, the hypotenuse is \(5\sqrt{2}\). Thus, the sine of 45 degrees can be calculated as:

$$ \sin(45^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} $$

Similarly, the cosine function, which is the ratio of the adjacent side to the hypotenuse, yields the same result:

$$ \cos(45^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} $$

The tangent function, defined as the ratio of the opposite side to the adjacent side, results in:

$$ \tan(45^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{5} = 1 $$

From these calculations, a clear pattern emerges: both the sine and cosine of 45 degrees equal \(\frac{\sqrt{2}}{2}\), while the tangent equals 1. This symmetry is a hallmark of the 45-45-90 triangle.

Next, we can explore the reciprocal trigonometric functions. The cosecant, which is the reciprocal of sine, is calculated as:

$$ \csc(45^\circ) = \frac{1}{\sin(45^\circ)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} $$

Similarly, the secant, the reciprocal of cosine, also results in:

$$ \sec(45^\circ) = \frac{1}{\cos(45^\circ)} = \sqrt{2} $$

Finally, the cotangent, the reciprocal of tangent, is:

$$ \cot(45^\circ) = \frac{1}{\tan(45^\circ)} = 1 $$

In summary, for the 45-45-90 triangle, the sine and cosine functions are consistently \(\frac{\sqrt{2}}{2}\), while the cosecant and secant are both \(\sqrt{2}\), and the tangent and cotangent are both 1. Recognizing these relationships can simplify the process of solving problems involving trigonometric functions in this special triangle.

30-60-90 Triangles

In the study of right triangles, the 30-60-90 triangle is a special case that frequently appears in various mathematical contexts, including geometry and physics. Understanding the relationships between the side lengths of this triangle is essential for solving problems efficiently.

In a 30-60-90 triangle, the side lengths are related to the shortest leg, which is opposite the 30-degree angle. The relationships can be summarized as follows: the hypotenuse is twice the length of the shortest leg, and the long leg (opposite the 60-degree angle) is equal to the shortest leg multiplied by the square root of 3. These relationships can be expressed mathematically as:

Hypotenuse \( = 2 \times \text{(short leg)} \)

Long leg \( = \text{(short leg)} \times \sqrt{3} \)

For example, if the shortest leg measures 5 units, the hypotenuse would be \( 5 \times 2 = 10 \) units, and the long leg would be \( 5 \times \sqrt{3} \) units.

To apply these relationships, it is crucial to identify the angles in the triangle. The angles in a triangle must sum to 180 degrees. Therefore, if one angle is 30 degrees and another is 90 degrees, the remaining angle must be 60 degrees, confirming that it is indeed a 30-60-90 triangle.

When given the hypotenuse, you can find the short leg by rearranging the relationship: divide the hypotenuse by 2. For instance, if the hypotenuse is 8 units, the short leg would be \( 8 / 2 = 4 \) units. Subsequently, the long leg can be calculated as \( 4 \times \sqrt{3} \) units.

Conversely, if the long leg is known, you can find the short leg by rearranging the relationship: divide the long leg by the square root of 3. For example, if the long leg measures 1 unit, the short leg would be \( 1 / \sqrt{3} \), which can be rationalized to \( \frac{\sqrt{3}}{3} \) units. The hypotenuse can then be determined by multiplying the short leg by 2, resulting in \( 2 \times \frac{\sqrt{3}}{3} \) units.

These shortcuts simplify the process of solving for unknown sides in 30-60-90 triangles, making it easier to tackle more complex problems in mathematics and related fields.

Given the triangle below, determine the missing side(s) without using the Pythagorean theorem (make sure your answer is fully simplified).

• A. $x=10,y=5\sqrt5$
• B. $x=3,y=4$
• C. $x=5\sqrt3,y=10$
• D. $x=5,y=5\sqrt2$

Answer: (C)

Without using a calculator, determine all values of A in the interval $\left[0,\frac{\pi}{2}\right)$ with the following trigonometric function value.
$\cos A=\frac{\sqrt3}{2}$

• A. $0$ only
• B. $\frac{\pi}{4}$ only
• C. $\frac{\pi}{6}$ only
• D. $\frac{\pi}{3}$ only

Answer: (C)

Common Trig Functions For 30-60-90 Triangles

In the study of special right triangles, particularly the 30-60-90 triangle, understanding the associated trigonometric functions is essential. The key angles in this triangle are 30 degrees and 60 degrees, and each has specific sine, cosine, and tangent values that can be derived from the triangle's side lengths.

For a 30-degree angle, the sine function is defined as the ratio of the length of the opposite side to the hypotenuse. In a 30-60-90 triangle, the opposite side to the 30-degree angle measures 5, while the hypotenuse measures 10. Thus, the sine of 30 degrees is calculated as:

$$\sin(30^\circ) = \frac{5}{10} = \frac{1}{2}$$

The cosine function, which is the ratio of the adjacent side to the hypotenuse, gives us:

$$\cos(30^\circ) = \frac{5\sqrt{3}}{10} = \frac{\sqrt{3}}{2}$$

For the tangent function, defined as the ratio of the opposite side to the adjacent side, we find:

$$\tan(30^\circ) = \frac{5}{5\sqrt{3}} = \frac{1}{\sqrt{3}}$$

Rationalizing the denominator results in:

$$\tan(30^\circ) = \frac{\sqrt{3}}{3}$$

Using reciprocal identities, we can derive the cosecant, secant, and cotangent functions. The cosecant is the reciprocal of sine:

$$\csc(30^\circ) = \frac{1}{\sin(30^\circ)} = 2$$

The secant is the reciprocal of cosine:

$$\sec(30^\circ) = \frac{1}{\cos(30^\circ)} = \frac{2}{\sqrt{3}}$$

Rationalizing gives:

$$\sec(30^\circ) = \frac{2\sqrt{3}}{3}$$

Finally, the cotangent is the reciprocal of tangent:

$$\cot(30^\circ) = \frac{1}{\tan(30^\circ)} = \sqrt{3}$$

For the 60-degree angle, the sine, cosine, and tangent functions can be derived similarly. The sine of 60 degrees is:

$$\sin(60^\circ) = \frac{5\sqrt{3}}{10} = \frac{\sqrt{3}}{2}$$

The cosine is:

$$\cos(60^\circ) = \frac{5}{10} = \frac{1}{2}$$

And the tangent is:

$$\tan(60^\circ) = \frac{5\sqrt{3}}{5} = \sqrt{3}$$

Using reciprocal identities again, we find:

$$\csc(60^\circ) = \frac{1}{\sin(60^\circ)} = \frac{2}{\sqrt{3}}$$

Rationalizing gives:

$$\csc(60^\circ) = \frac{2\sqrt{3}}{3}$$

For secant:

$$\sec(60^\circ) = \frac{1}{\cos(60^\circ)} = 2$$

And for cotangent:

$$\cot(60^\circ) = \frac{1}{\tan(60^\circ)} = \frac{1}{\sqrt{3}}$$

Rationalizing gives:

$$\cot(60^\circ) = \frac{\sqrt{3}}{3}$$

Notably, patterns emerge in these trigonometric functions. The sine and cosine values for 30 degrees and 60 degrees switch places, as do the cosecant and secant values. Similarly, the tangent and cotangent values exhibit a similar switching behavior. Recognizing these patterns can simplify the process of calculating trigonometric functions for special right triangles.

Here’s what students ask on this topic:

What are the properties of a 45-45-90 triangle?

A 45-45-90 triangle is a special right triangle where the two legs are of equal length, and the angles are 45°, 45°, and 90°. The hypotenuse is √2 times the length of each leg. This triangle is isosceles, meaning it has two equal sides and two equal angles. The consistent relationship between the sides allows for quick calculations without using the Pythagorean theorem. The trigonometric functions for this triangle are also straightforward: sine and cosine of 45° are √2/2, and tangent is 1. These properties make the 45-45-90 triangle useful in various mathematical and scientific applications.

How do you solve a 30-60-90 triangle?

To solve a 30-60-90 triangle, use the relationships between the side lengths: the hypotenuse is twice the length of the shortest leg, and the longer leg is √3 times the shortest leg. If the shortest leg is known, multiply it by 2 to find the hypotenuse and by √3 to find the longer leg. These shortcuts simplify calculations and avoid using the Pythagorean theorem. The trigonometric functions for the 30° and 60° angles also follow patterns: sine of 30° is 1/2, cosine is √3/2, and tangent is √3/3. Understanding these relationships helps in solving problems efficiently.

What are the trigonometric functions for a 45-45-90 triangle?

In a 45-45-90 triangle, the trigonometric functions are consistent due to the equal angles. The sine and cosine of 45° are both √2/2, while the tangent is 1. The reciprocal identities are straightforward: cosecant and secant are √2, and cotangent is 1. These functions are derived from the equal leg lengths and the hypotenuse being √2 times a leg. This symmetry simplifies calculations and is useful in various applications, including geometry and trigonometry problems. Recognizing these patterns aids in solving problems quickly and accurately.

How do trigonometric values switch in a 30-60-90 triangle?

In a 30-60-90 triangle, the trigonometric values switch between the 30° and 60° angles. For the 30° angle, sine is 1/2, cosine is √3/2, and tangent is √3/3. For the 60° angle, sine becomes √3/2, cosine is 1/2, and tangent is √3. This switch occurs because the opposite and adjacent sides change relative to the angle being considered. The reciprocal identities also switch: cosecant and secant values swap between the angles. Understanding these switches helps in recognizing patterns and solving problems efficiently, especially in trigonometry and geometry.

What are the shortcuts for solving special right triangles?

Special right triangles, such as 45-45-90 and 30-60-90 triangles, have shortcuts for solving them quickly. In a 45-45-90 triangle, the hypotenuse is √2 times the length of a leg. In a 30-60-90 triangle, the hypotenuse is twice the shortest leg, and the longer leg is √3 times the shortest leg. These shortcuts eliminate the need for the Pythagorean theorem and simplify calculations. Recognizing these patterns allows for efficient problem-solving in geometry and trigonometry, making these triangles useful in various mathematical and scientific contexts.