Question

trigonometric ratios instruction
the three basic trigonometric functions for the acute angles of a right triangle are sine, cosine, and tangent. the functions relate a given angle measure and a ratio of side lengths.
· $\tan(a)=\frac{\underline{\quad\quad}}{\text{adjacent}}=\frac{bc}{\underline{\quad\quad}}$
· $\underline{\quad\quad}$ of angle a = $\tan(a)$
trigonometric ratios
· $\sin(b)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{\underline{\quad\quad}}{ab}$
· $\cos(b)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{\underline{\quad\quad}}{ab}$
· $\tan(b)=\frac{\text{opposite}}{\underline{\quad\quad}}=\frac{ac}{bc}$
the $\underline{\quad\quad}$ angle of the right triangle does not have trigonometric ratios, because the side opposite of the right angle is the hypotenuse, and both $\underline{\quad\quad}$ are adjacent to the right angle.

Explanation:

Step1: Analyze the first tangent formula for angle A

In a right triangle, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side. For angle A in the first triangle (right - angled at C), the side opposite to angle A is BC, and the side adjacent to angle A is AC. So, \(\tan(A)=\frac{\text{opposite}}{\text{adjacent}}=\frac{BC}{AC}\). Also, the tangent of an angle is also called the tangent ratio, so the blank for "______ of angle A = tan(A)" should be "tangent ratio".

Step2: Analyze the sine formula for angle B

For angle B in the right - angled triangle at C, the side opposite to angle B is AC (because in triangle ABC, right - angled at C, angle B is at vertex B, so the side opposite to B is AC) and the hypotenuse is AB. So, \(\sin(B)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{AC}{AB}\).

Step3: Analyze the cosine formula for angle B

The side adjacent to angle B is BC (in triangle ABC, right - angled at C, angle B is at vertex B, so the side adjacent to B is BC) and the hypotenuse is AB. So, \(\cos(B)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{BC}{AB}\).

Step4: Analyze the tangent formula for angle B

The tangent of an angle is the ratio of the opposite side to the adjacent side. For angle B, the opposite side is AC and the adjacent side is BC. So, \(\tan(B)=\frac{\text{opposite}}{\text{adjacent}}=\frac{AC}{BC}\).

Step5: Analyze the last part about the angle without trigonometric ratios

In a right - triangle, the right angle (the \(90^{\circ}\) angle) does not have trigonometric ratios in the same way as acute angles. The two sides adjacent to the right angle are the legs of the right - triangle. The right angle's opposite side is the hypotenuse, and the definitions of sine, cosine, and tangent (which require a non - right acute angle) do not apply to the right angle. So, the first blank is "right" and the second blank is "legs".

Answer:

• For \(\tan(A)=\frac{\text{opposite}}{\text{adjacent}}=\frac{BC}{AC}\) (first \(\tan(A)\) formula), "opposite" and "AC" fill the blanks. The blank for "______ of angle A = tan(A)" is "tangent ratio".
• For \(\sin(B)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{AC}{AB}\), "AC" fills the blank.
• For \(\cos(B)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{BC}{AB}\), "BC" fills the blank.
• For \(\tan(B)=\frac{\text{opposite}}{\text{adjacent}}=\frac{AC}{BC}\), "adjacent" fills the blank.
• For "The ____ angle of the right triangle does not have trigonometric ratios, because the side opposite of the right angle is the hypotenuse, and both ____ are adjacent to the right angle", the first blank is "right" and the second blank is "legs".