Hãy tính sinα và tgα, nếu:
a) \(\cos \alpha = {5 \over {13}}\);
b) \(\cos \alpha = {{15} \over {17}}\);
c) \(\cos \alpha = 0,6.\)
Gợi ý làm bài
a) \(cos \alpha = {5 \over {13}}\)
* Ta có:
\({\sin ^2}\alpha + {\cos ^2}\alpha = 1\)
Suy ra:
\(\eqalign{
& {\sin ^2}\alpha = 1 - {\cos ^2}\alpha = 1 - {\left( {{5 \over {13}}} \right)^2} \cr
& = 1 - {{25} \over {169}} = {{144} \over {169}} \cr} \)
Vì \(\sin \alpha > 0\) nên \(\sin \alpha = \sqrt {{{144} \over {169}}} = {{12} \over {13}}\)
* \(tg\alpha = {{\sin \alpha } \over {\cos \alpha }} = {{{{12} \over {13}}} \over {{5 \over {13}}}} = {{12} \over {13}}.{{13} \over 5} = {{12} \over 5}\)
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b) \(\cos \alpha = {{15} \over {17}}\)
* Ta có: \({\sin ^2}\alpha + {\cos ^2}\alpha = 1\)
Suy ra:
\(\eqalign{
& {\sin ^2}\alpha = 1 - {\cos ^2}\alpha = 1 - {\left( {{{15} \over {17}}} \right)^2} \cr
& = 1 - {{225} \over {289}} = {{64} \over {289}} \cr} \)
Vì \(\sin \alpha > 0\) nên \(\sin \alpha = \sqrt {{{64} \over {289}}} = {8 \over {17}}\)
* \(tg\alpha {{\sin \alpha } \over {\cos \alpha }} = {{{8 \over {17}}} \over {{{15} \over {17}}}} = {8 \over {17}}.{{17} \over {15}} = {8 \over {15}}\)
c) \(\cos \alpha = 0,6\)
* Ta có: \({\sin ^2}\alpha + {\cos ^2}\alpha = 1.\)
Suy ra: \({\sin ^2}\alpha = 1 - {\cos ^2}\alpha \)
\( = 1 - {(0,6)^2} = 1 - 0,36 = 0,64\)
Vì \(\sin \alpha > 0\) nên \(\sin \alpha = \sqrt {0,64} = 0,8\)
* \(tg\alpha = {{\sin \alpha } \over {\cos \alpha }} = {{0,8} \over {0,6}} = {8 \over 6} = {4 \over 3}\)