Page 2 of a trigonometry worksheet featuring exercises on calculating trigonometric ratios for various right triangles and angles.
Math worksheet with problems 11-24 asking students to find trigonometric ratios like sine, cosine, and tangent for right triangles.
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Step-by-step solution for: Trig. Ratios Practice worksheet
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Show Answer Key & Explanations
Step-by-step solution for: Trig. Ratios Practice worksheet
Here are the step-by-step solutions for each problem on the worksheet.
11) cos Z
* Identify sides: In triangle $XYZ$, angle $Y$ is the right angle ($90^\circ$). We are looking at angle $Z$.
* The side next to angle $Z$ (Adjacent) is $YZ = 12$.
* The longest side opposite the right angle (Hypotenuse) is $XZ = 15$.
* Formula: $\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Calculate: $\cos Z = \frac{12}{15} = 0.8$
* Round: To the nearest ten-thousandth, this is 0.8000.
12) cos C
* Identify sides: In triangle $ABC$, angle $B$ is the right angle. We are looking at angle $C$.
* Adjacent side ($BC$) = 27.
* Hypotenuse ($AC$) = 45.
* Formula: $\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Calculate: $\cos C = \frac{27}{45} = 0.6$
* Round: 0.6000.
13) tan C
* Identify sides: In triangle $ABC$, angle $B$ is the right angle. We are looking at angle $C$.
* Side opposite angle $C$ ($AB$) = 40.
* Side adjacent to angle $C$ ($BC$) = 30.
* Formula: $\tan = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculate: $\tan C = \frac{40}{30} \approx 1.33333...$
* Round: 1.3333.
14) tan A
* Identify sides: In triangle $ABC$, angle $B$ is the right angle. We are looking at angle $A$.
* Side opposite angle $A$ ($BC$) = 21.
* Side adjacent to angle $A$ ($AB$) = 20.
* Formula: $\tan = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculate: $\tan A = \frac{21}{20} = 1.05$
* Round: 1.0500.
15) tan C
* Identify sides: In triangle $ABC$, angle $B$ is the right angle. We are looking at angle $C$.
* Side opposite angle $C$ ($AB$) = 12.
* Side adjacent to angle $C$ ($BC$) = 35.
* Formula: $\tan = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculate: $\tan C = \frac{12}{35} \approx 0.342857...$
* Round: The fifth decimal digit is 5, so we round up. 0.3429.
16) tan X
* Identify sides: In triangle $XYZ$, angle $Y$ is the right angle. We are looking at angle $X$.
* Side opposite angle $X$ ($YZ$) = 30.
* Side adjacent to angle $X$ ($XY$) = 40.
* Formula: $\tan = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculate: $\tan X = \frac{30}{40} = 0.75$
* Round: 0.7500.
17) sin Z
* Identify sides: In triangle $XYZ$, angle $Y$ is the right angle. We are looking at angle $Z$.
* Side opposite angle $Z$ ($XY$) = 12.
* Hypotenuse ($XZ$) = 37.
* Formula: $\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Calculate: $\sin Z = \frac{12}{37} \approx 0.324324...$
* Round: 0.3243.
18) sin Z
* Identify sides: In triangle $XYZ$, angle $Y$ is the right angle. We are looking at angle $Z$.
* Side opposite angle $Z$ ($XY$) = 30.
* Hypotenuse ($XZ$) = 50.
* Formula: $\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Calculate: $\sin Z = \frac{30}{50} = 0.6$
* Round: 0.6000.
19) sin 48°
* Use a calculator set to Degree mode.
* $\sin(48) \approx 0.743144...$
* Round: 0.7431.
20) sin 38°
* Use a calculator.
* $\sin(38) \approx 0.615661...$
* Round: 0.6157.
21) cos 61°
* Use a calculator.
* $\cos(61) \approx 0.484809...$
* Round: 0.4848.
22) cos 51°
* Use a calculator.
* $\cos(51) \approx 0.629320...$
* Round: 0.6293.
23) Can the sine of an angle ever equal 2? Why or why not?
* Reasoning: The sine ratio is defined as $\frac{\text{Opposite}}{\text{Hypotenuse}}$. In a right-angled triangle, the hypotenuse is always the longest side. Therefore, the numerator (Opposite) will always be smaller than or equal to the denominator (Hypotenuse). This means the result can never be greater than 1.
* Answer: No. The value of sine must always be between -1 and 1. Since the hypotenuse is the longest side of a right triangle, the ratio $\frac{\text{opposite}}{\text{hypotenuse}}$ cannot exceed 1.
24) $\sin x = \frac{1}{3}$. Find $\cos x$.
* Step 1: Imagine a right triangle where angle is $x$. Since $\sin x = \frac{1}{3}$, let the Opposite side be $1$ and the Hypotenuse be $3$.
* Step 2: Use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the Adjacent side ($b$).
$$1^2 + b^2 = 3^2$$
$$1 + b^2 = 9$$
$$b^2 = 8$$
$$b = \sqrt{8}$$
* Step 3: Calculate $\cos x$. The formula is $\frac{\text{Adjacent}}{\text{Hypotenuse}}$.
$$\cos x = \frac{\sqrt{8}}{3}$$
* Step 4: Convert to a decimal.
$$\sqrt{8} \approx 2.828427$$
$$\frac{2.828427}{3} \approx 0.942809...$$
* Round: 0.9428.
Final Answer:
11) 0.8000
12) 0.6000
13) 1.3333
14) 1.0500
15) 0.3429
16) 0.7500
17) 0.3243
18) 0.6000
19) 0.7431
20) 0.6157
21) 0.4848
22) 0.6293
23) No, because the hypotenuse is always the longest side, so the ratio cannot be greater than 1.
24) 0.9428
11) cos Z
* Identify sides: In triangle $XYZ$, angle $Y$ is the right angle ($90^\circ$). We are looking at angle $Z$.
* The side next to angle $Z$ (Adjacent) is $YZ = 12$.
* The longest side opposite the right angle (Hypotenuse) is $XZ = 15$.
* Formula: $\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Calculate: $\cos Z = \frac{12}{15} = 0.8$
* Round: To the nearest ten-thousandth, this is 0.8000.
12) cos C
* Identify sides: In triangle $ABC$, angle $B$ is the right angle. We are looking at angle $C$.
* Adjacent side ($BC$) = 27.
* Hypotenuse ($AC$) = 45.
* Formula: $\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Calculate: $\cos C = \frac{27}{45} = 0.6$
* Round: 0.6000.
13) tan C
* Identify sides: In triangle $ABC$, angle $B$ is the right angle. We are looking at angle $C$.
* Side opposite angle $C$ ($AB$) = 40.
* Side adjacent to angle $C$ ($BC$) = 30.
* Formula: $\tan = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculate: $\tan C = \frac{40}{30} \approx 1.33333...$
* Round: 1.3333.
14) tan A
* Identify sides: In triangle $ABC$, angle $B$ is the right angle. We are looking at angle $A$.
* Side opposite angle $A$ ($BC$) = 21.
* Side adjacent to angle $A$ ($AB$) = 20.
* Formula: $\tan = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculate: $\tan A = \frac{21}{20} = 1.05$
* Round: 1.0500.
15) tan C
* Identify sides: In triangle $ABC$, angle $B$ is the right angle. We are looking at angle $C$.
* Side opposite angle $C$ ($AB$) = 12.
* Side adjacent to angle $C$ ($BC$) = 35.
* Formula: $\tan = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculate: $\tan C = \frac{12}{35} \approx 0.342857...$
* Round: The fifth decimal digit is 5, so we round up. 0.3429.
16) tan X
* Identify sides: In triangle $XYZ$, angle $Y$ is the right angle. We are looking at angle $X$.
* Side opposite angle $X$ ($YZ$) = 30.
* Side adjacent to angle $X$ ($XY$) = 40.
* Formula: $\tan = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculate: $\tan X = \frac{30}{40} = 0.75$
* Round: 0.7500.
17) sin Z
* Identify sides: In triangle $XYZ$, angle $Y$ is the right angle. We are looking at angle $Z$.
* Side opposite angle $Z$ ($XY$) = 12.
* Hypotenuse ($XZ$) = 37.
* Formula: $\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Calculate: $\sin Z = \frac{12}{37} \approx 0.324324...$
* Round: 0.3243.
18) sin Z
* Identify sides: In triangle $XYZ$, angle $Y$ is the right angle. We are looking at angle $Z$.
* Side opposite angle $Z$ ($XY$) = 30.
* Hypotenuse ($XZ$) = 50.
* Formula: $\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Calculate: $\sin Z = \frac{30}{50} = 0.6$
* Round: 0.6000.
19) sin 48°
* Use a calculator set to Degree mode.
* $\sin(48) \approx 0.743144...$
* Round: 0.7431.
20) sin 38°
* Use a calculator.
* $\sin(38) \approx 0.615661...$
* Round: 0.6157.
21) cos 61°
* Use a calculator.
* $\cos(61) \approx 0.484809...$
* Round: 0.4848.
22) cos 51°
* Use a calculator.
* $\cos(51) \approx 0.629320...$
* Round: 0.6293.
23) Can the sine of an angle ever equal 2? Why or why not?
* Reasoning: The sine ratio is defined as $\frac{\text{Opposite}}{\text{Hypotenuse}}$. In a right-angled triangle, the hypotenuse is always the longest side. Therefore, the numerator (Opposite) will always be smaller than or equal to the denominator (Hypotenuse). This means the result can never be greater than 1.
* Answer: No. The value of sine must always be between -1 and 1. Since the hypotenuse is the longest side of a right triangle, the ratio $\frac{\text{opposite}}{\text{hypotenuse}}$ cannot exceed 1.
24) $\sin x = \frac{1}{3}$. Find $\cos x$.
* Step 1: Imagine a right triangle where angle is $x$. Since $\sin x = \frac{1}{3}$, let the Opposite side be $1$ and the Hypotenuse be $3$.
* Step 2: Use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the Adjacent side ($b$).
$$1^2 + b^2 = 3^2$$
$$1 + b^2 = 9$$
$$b^2 = 8$$
$$b = \sqrt{8}$$
* Step 3: Calculate $\cos x$. The formula is $\frac{\text{Adjacent}}{\text{Hypotenuse}}$.
$$\cos x = \frac{\sqrt{8}}{3}$$
* Step 4: Convert to a decimal.
$$\sqrt{8} \approx 2.828427$$
$$\frac{2.828427}{3} \approx 0.942809...$$
* Round: 0.9428.
Final Answer:
11) 0.8000
12) 0.6000
13) 1.3333
14) 1.0500
15) 0.3429
16) 0.7500
17) 0.3243
18) 0.6000
19) 0.7431
20) 0.6157
21) 0.4848
22) 0.6293
23) No, because the hypotenuse is always the longest side, so the ratio cannot be greater than 1.
24) 0.9428
Parent Tip: Review the logic above to help your child master the concept of trig ratios practice worksheet.
