Abstract
The study of trigonometry has posed significant challenges to students over many years because of the lack of appropriate models that can be used to find required trigonometric ratios. The traditional approach to finding such answers has been the use of a right-angled triangle. Here, learners use the SOHCAHTOA formula to calculate angles in a triangle and the length of all of its sides. However, the advent of new mathematics brought about the use of the unit circle, which is a model that enables learners to master the trigonometric ratio for angles inside the circle. Notably, students have been unable to identify the effect of changing the radius of the unit circle on the angles and trigonometric ratios. Thus, unit cancellation, which entails eliminating the square of sines and cosines sometimes leads to wrong answers. However, the calculations in this paper show that increasing the radius of a unit circle does not affect the angles and trigonometric angles in the circle in any way.
Introduction
A unit circle is an important approach that is used by mathematicians to make various measures of angles and determine trigonometric ratios. Normally, it is assumed that the outcomes are pegged on the fact that the radius of the circle is 1 (Muniz, n.d). Thus, there have been questions on the impact that changing the length of the radius would have on the angles inside the triangle and the trigonometric ratios. In such a way, a need arises for mathematicians to understand how increasing or reducing the radius of a circle affects the angles subtended at the center of the circle and trigonometric ratios. This paper seeks to use the radius of the circle as 2 and use this value to create a right-angled triangle inside the circle. Thereafter the measurements in the expanded circle will be taken with the angle remaining the same. The lengths of the two sides will be used to determine various trigonometric ratios. Notably, increasing the length of the radius means that the adjacent of the angle subtended at the center doubles. The hypotenuse also increases by its length. After that, with two separate circles of radius 1 and 2 respectively, the angle subtended at the center will be increased in such a way that it is the same for the two circles. Using trigonometric ratios, the two angles will be determined. Finally, a comparison will be made to evaluate whether there exists any significant change in the angle subtended at the center of a unit circle if the radius is doubled, or if this action impacts the trigonometric ratios inside the circle.
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Background
Mickey and McClelland (2017) address the fact that high school students have been finding it hard to understand the topic of trigonometry, which has jeopardized their chances of qualifying for careers in mathematics, engineering, and technology. In such a way, the authors emphasize the importance of the unit circle as a model that simplifies the understanding of the aforementioned topic. Moore, Laforest, and Kim (2010) on the other hand posit that there is a lack of a clear connection between the understandings of teachers and students in trigonometry and the unit circle. In their work, they explore a teaching experiment in which the notion of various mathematics teachers is collected. From the research, it was noted that students experienced a significant challenge when there was no clear indication that the radius of the circles given to them were one. Thus, the learners are forced to use unit cancellation as a way of relating the circles provided to the unit circle.
Kendal and Stacey seek to compare the ratio method that was initially used in the understanding of the topic of trigonometry. In their exploration, they allocated eight classes of students randomly to either method of teaching trigonometry to evaluate their effectiveness. According to them, the ratio method turned out to be the most appropriate method as it led to more positive results. In such a way, they advocated for the use of this method for an easy understanding of trigonometry. Weber analyzed two college courses to estimate the level of students’ understanding of trigonometry. In their exploration, they compared learners in a lecture-based class with those in a class taught by a specialist in theories of learning.
The exploration found out that the students who were exposed to experimental instruction developed a deeper understanding of the topic.
Problem Description and Method
The unit circle, which has a radius of 1, has been instrumental in helping learners to understand trigonometry. Notably, it has made it easy for them to determine various angles in the circle and find various trigonometric ratios. However, learners have experienced significant challenges when presented with circles where the radius are not indicated clearly. In such a way, they are forced to engage in unit cancellation to find the right solutions for the problems. In such a way, this paper seeks to outline the possible effects of adjusting the radius of the unit circle on the measurements aforementioned.
The method followed in this exercise entails drawing two circles, one inside another. The inner circle has a radius of 1, whereas the outer circle has a radius of 2. Using a common angle of 62.57 o , and the known radius, the horizontal and vertical sides of the triangles inside the circles can be calculated. Importantly, these lengths can be used to determine the coordinates of the point of intersection of the radii and the circumferences. Therefore, the trigonometric ratios for the second triangle can be determined and compared to those for the first one. In such a way, the debate on the effect of the changes in the radius of the unit circle can be assessed effectively to alleviate the confusion that has existed over a long time.
Findings
From the circles above, a few findings are clear. To start with, the angle subtended at the centers of the two circles is the same (62.57 o ). Using this value, it can be seen that the coordinates of the point of intersection of the radius and the circumference of the first circle are (0.46, 0.89). From these, 0.46 is the side adjacent to the angle, whereas 0.89 is the side opposite to the angle. In addition, the radius of the circle is 1.
The cosine of 62.57 for the smaller circle can be calculated by 0.46/1= 0.46
The sine of 62.57 for the smaller circle can be calculated by 0.89/1= 0.89
The tangent of 62.57 for the smaller circle can be calculated by 0.89/0.46= 1.934
These ratios can also be calculated for the bigger circle in which the angle subtended at the center of the circle is still 62.57. The coordinates of the point of intersection of the radius and the circumference of the circle are (0.92, 1.78). In such a way, the trigonometric ratios for the circle can be calculated as follows;
The cosine of 62.57 in the bigger circle = 0.92/2= 0.46
The sine of 62.57 in the bigger circle = 1.78/2= 0.89
The tangent of 62.57 in the bigger circle is 1.78/0.92= 1.934
In such a way, there is no significant change in the trigonometric ratios of a unit circle, regardless of changes in the length of the radius.
It is also worth noting that changing the angle subtended at the center of the circles of radius 1 and 2 respectively will not have any effect on the trigonometric ratios, even though the coordinates of the points of intersection of the radii and the circumferences of the circles would change. For instance, the cosine of 30o for a circle of radius 1 would be 0.8660. This implies that the length of the adjacent side of the triangle formed inside the circle would be 0.8660. The value of the sine of the same angle is 0.5, meaning that they-coordinate for the point of intersection of the radius and the circumference will be 1. Therefore, the tangent for the angle will be 0.5/0.8660= 0.5773. If the radius of the circle is increased to 2, the adjacent side also increases to 1.732. Therefore;
The cosine of 30 for the bigger circle will be 1.732/2=0.8660
The sine of 30 for the bigger circle will be ½= 0.5
The tangent of 30 for the bigger angle will be 1/1.732= 0.5773
Conclusion
In conclusion, this paper has explored the challenges that students have faced when solving questions in trigonometry. The absence of an appropriate model left learners only with the right-angled triangle, which was a tedious method of finding answers to the questions at hand. However, the introduction of the unit circle has made calculations easier. As shown in the paper, changing the radius of the unit circle does not affect measurements and trigonometric ratios in a circle.
References
Kendal, M., & Stacey, K. (n.d.). Trigonometry: Comparing Ratio and Unit Circle Methods. Retrieved from https://pdfs.semanticscholar.org/bb94/fd29c031628ec911645d6ebf68c480ead42d.pdf.
Mickey, K. W., & McClelland, J. L. (2017). The Unit Circle as a Grounded Conceptual Structure in Precalculus Trigonometry. Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts , 247-269.
Moore, K. C., Laforest, K., & Kim, H. J. (2010). Unit Conversions. The Biodiesel Handbook , 479-481. doi:10.1016/b978-1-893997-62-2.50019-x.
Muniz, H. (n.d.). 3 Expert Tips for Using the Unit Circle. Retrieved from https://blog.prepscholar.com/unit-circle-chart-radians-degrees.
Weber, K. (2005). Students’ understanding of trigonometric functions. Mathematics Education Research Journal , 17 (3), 91-112. doi:10.1007/bf03217423.
