Question

Verify each step in parts a through e. Then solve parts f and g. a. \(\sin P=\frac{p}{q}\) and \(\cos P=\frac{r}{q}\) b. \(\sin ^{2} P=\frac{p^{2}}{q^{2}}\) and \(\cos ^{2} P=\frac{r^{2}}{q^{2}}\) C. \(\sin ^{2} P+\cos ^{2} P=\frac{p^{2}}{q^{2}}+\frac{r^{2}}{q^{2}}\) or \(\frac{p^{2}+r^{2}}{q^{2}}\) d. \(p^{2}+r^{2}=q^{2}\) e. \(\sin ^{2} P+\cos ^{2} P=\frac{q^{2}}{q^{2}}\) or 1 f. Find \(\sin x\) if \(\cos x=\frac{3}{5}\). g. Find \(\cos x\) if \(\sin x=\frac{5}{13}\).

Short answer

f. \(\sin x = \pm \frac{4}{5}\); g. \(\cos x = \pm \frac{12}{13}\).

Step-by-step solution

Verify Part a

Given: \(\sin P=\frac{p}{q}\) and \(\cos P=\frac{r}{q}\). These are the definitions of sine and cosine for angle \(P\) in a right triangle where \(p\) is the opposite side, \(q\) is the hypotenuse, and \(r\) is the adjacent side.

Verify Part b

Compute \(\sin^2 P\) and \(\cos^2 P\):- \(\sin^2 P = \left(\frac{p}{q}\right)^2 = \frac{p^2}{q^2}\)- \(\cos^2 P = \left(\frac{r}{q}\right)^2 = \frac{r^2}{q^2}\)

Verify Part c

Add the squares of \(\sin P\) and \(\cos P\):- \(\sin^2 P + \cos^2 P = \frac{p^2}{q^2} + \frac{r^2}{q^2} = \frac{p^2 + r^2}{q^2}\)

Verify Part d

We are given that \(p^2 + r^2 = q^2\), which follows from the Pythagorean theorem for a right triangle.

Verify Part e

The expression \(\sin^2 P + \cos^2 P = \frac{q^2}{q^2} = 1\) stems from the identity, confirming that \(\sin^2 P + \cos^2 P = 1\).

Solve Part f

Given \(\cos x = \frac{3}{5}\), use the Pythagorean identity:\(\sin^2 x + \cos^2 x = 1\).Substitute for \(\cos x\):\(\sin^2 x + \left(\frac{3}{5}\right)^2 = 1\).Simplify:\(\sin^2 x + \frac{9}{25} = 1\).Subtract \(\frac{9}{25}\):\(\sin^2 x = 1 - \frac{9}{25} = \frac{16}{25}\).Take the square root:\(\sin x = \pm \frac{4}{5}\). Depending on the quadrant, \(\sin x\) can be either positive or negative.

Solve Part g

Given \(\sin x = \frac{5}{13}\), use the Pythagorean identity:\(\sin^2 x + \cos^2 x = 1\).Substitute for \(\sin x\):\(\left(\frac{5}{13}\right)^2 + \cos^2 x = 1\).Simplify:\(\frac{25}{169} + \cos^2 x = 1\).Subtract \(\frac{25}{169}\):\(\cos^2 x = 1 - \frac{25}{169} = \frac{144}{169}\).Take the square root:\(\cos x = \pm \frac{12}{13}\). Depending on the quadrant, \(\cos x\) can be either positive or negative.

Key concepts

Right Triangle Trigonometry

Right triangle trigonometry involves understanding the relationships between the angles and sides of a right triangle. In such triangles, one angle is always 90 degrees. The sides adjacent to this angle are especially important. The side opposite the angle of interest is called the "opposite," the side next to the angle but not the hypotenuse is the "adjacent," and the longest side is the "hypotenuse."

The primary trigonometric functions are sine, cosine, and tangent. - Sine (\(\sin\theta\)) is the ratio of the opposite side to the hypotenuse.- Cosine (\(\cos\theta\)) is the ratio of the adjacent side to the hypotenuse.- Tangent (\(\tan\theta\)) is the ratio of the opposite side to the adjacent side.
These functions allow us to solve for unknown angles or sides in right triangles. To verify such relationships, one can use trigonometric identities, particularly the Pythagorean identity, which links the sine and cosine of an angle.

Pythagorean Theorem

The Pythagorean Theorem is a central concept in geometry that applies to right triangles. It states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is represented as: \[c^2 = a^2 + b^2\]Where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.

This theorem is not only useful for verifying the relationship between the sides of a triangle but is also vital in deriving trigonometric identities. For example, the identity \(\sin^2\theta + \cos^2\theta = 1\) can be derived from the Pythagorean Theorem when applied to a unit circle. This forms the foundation for many trigonometric proofs, making it instrumental for solving exercises that require verification of trigonometric identities.

Sine and Cosine Functions

The sine and cosine functions are pivotal in trigonometry, often used to explore relationships within a right triangle. These functions are ratios formed based on the angles of the triangle.

Understanding these functions involves knowing that:

• The sine function, \(\sin\theta\), is calculated as the ratio of the length of the opposite side to the hypotenuse.
• Cosine, \(\cos\theta\), is the ratio of the adjacent side to the hypotenuse.

These ratios are constant for a given angle, irrespective of the size of the triangle, demonstrating the universal nature of these trigonometric relationships.

A vital identity for sine and cosine is \(\sin^2\theta + \cos^2\theta = 1\).This identity states that when the squares of sine and cosine of an angle are added together, the result is always 1 for any angle in a right triangle. This fundamental identity is useful for checking the correctness of trigonometric computations and is a key element in solving trigonometric problems.